Situation:
Level of students: 6th grade
Number of Students: 29 students
Grouping Arrangement: groups of 2 – 4
Previous Experience: Students have already completed Investigation I in Prime Time Factors and Multiples in the Connected Math series. Students have also reviewed divisibility rules.
Math Core Content Addressed:
MA-M-1.1.1 Rational Numbers
MA-M-1.2.4 Identify and use number theory concepts
Outcomes:
Students will classify numbers as abundant, deficient, or perfect based on the relationship between the number and its factors.
Materials:
Divisibility rules
Calculators
Investigating Math with Calculators in the Middle Grades:
Transparency Pages 22
Worksheet Pages 23
Prime Time
Completed lab sheet 1.2 and transparency (the factor game)
Prime factor list of primes up to 1000
Introduction:
Review factor game using the completed lab sheet. When playing the factor game which type of numbers where the best for you to choose? Which type of numbers were the worst for you to choose?
Discuss possible strategies for determining all of the proper factors of a given number:
a. divisibility rules
b. calculator-Explorer Plus- Write the number as a fraction using the number as both the numerator and the denominator. Use the SIMP key and divide by 2 if your number is an even number, and divide by an odd prime number if the number is odd; then use the ENTER key. If the number can be divided by the factor chosen it will simplify to a smaller number. If it can’t be divided by that number, it will not change. List the factors you can simplify the number by. Continue dividing numbers until the display reads 1/1. The proper factors will be the numbers you listed and any combination of the numbers that can be multiplied together and give you a number less than the original. These factors can be checked by dividing them individually into the original number. ( Ex. 12 Type 12/12 SIMP 2 ENTER Display reads 6/6. SIMP 2 Enter Display reads 3/3 SIMP 3 ENTER display reads 1/1. The factors are 1, 2, 2, 3 2*2=4 & 2*3=6. Therefore the proper factors of 12 are 1, 2, 3, 4, 6. Add these digits to get their sum
c. List the factors
d. Prime factorization
e. Introduce vocabulary: perfect numbers, abundant numbers, and deficient numbers
Look at completed lab sheet 1.2 to compare opponent’s score and your score to determine whether the number is perfect, abundant, or deficient. What kinds of numbers are always deficient?
Working in your groups of 2-4, use your calculators for determining all of the factors for the two, three, and four digit numbers on worksheet 23.
Find and record the proper factors
Record the sum of the factors and record whether the number is abundant, deficient, or perfect.
When you are finished identifying the numbers, as a group talk and record any patterns or observations you can make from this information.
Closure:
As a whole group share observations and record them on the overhead.
Assessment:
As a class, check the last two columns of the worksheet. On the back of their worksheet, have each student individually determine if 324 is a perfect, abundant or deficient number. They turn in their worksheet when finished.
Calculator Rationale:
Saves time finding factors for larger numbers
Prime Factors
Number theory has fascinated mathematicians for years. Fundamental to number theory are numbers themselves, and the basic building blocks for numbers are prime numbers. A prime number is a counting number that only has two factors, itself and one. Counting numbers which have more than two factors (such as six, whose factors are 1, 2, 3 and 6), are said to be composite numbers. The number one only has one factor and is considered to be neither prime nor composite.
When a composite number is written as a product of all of its prime factors, we have the prime factorization of the number. For example, the number 72 can be written as a product of primes as: 72 =. The expression "" is said to be the prime factorization of 72. The Fundamental Theorem of Arithmetic states that every composite number can be factored uniquely (except for the order of the factors) into a product of prime factors. What this means is that how you choose to factor a number into prime factors makes no difference. When you are done, the prime factorizations are essentially the same. Examine the two factor trees for 72 given below.
When we get done factoring using either set of factors to start with, we still have three factors of two and two factors of three or . This would be true if we had started to factor 72 as 24 times 3, 4 times 18, or any other pair of factors for 72.
Knowing the rules for divisibility will be very helpful when seeking to write a number in prime factorization form. Since a number is divisible by two if it ends in either 0, 2, 4, 6, or 8, it should be noted that two is the only even prime number. Another way to factor a number other than using factor trees is to start dividing by prime numbers, as shown below.
Once again, we can see that 72 =. Another key idea in writing the prime factorization of a number is an understanding of exponents. An exponent tells how many times the base is used as a factor. In the prime factorization of 72 =, the two is used as a factor three times and the three is used as a factor twice.
When checking to see if a number is prime or not, you need only divide by those prime numbers which when squared remain less than the given number. For example to see if 131 is prime, you need only check for divisibility by 2, 3, 5, 7, and 11, since 132 = 169. If a prime number greater than 13 divided 131, then the other factor would have to be less than 13 and you would have checked those already.
Introducing the Concept
Developing the Concept
What is an integer? { ... -3, -2, -1, 0, 1, 2, 3, ... }
Integers are the whole numbers, negative whole numbers, and zero. For example, 43434235, 28, 2, 0, -28, and -3030 are integers, but numbers like 1/2, 4.00032, 2.5, Pi, and -9.90 are not. We can say that an integer is in the set: {...3,-2,-1,0,1,2,3,...} (the three dots mean you keep going in both directions.)
It is often useful to think of the integers as points along a 'number line', like this:
Note that zero is neither positive nor negative.
About integers
The terms even and odd only apply to integers; 2.5 is neither even nor odd. Zero, on the other hand, is even since it is 2 times some integer: it's 2 times 0. To check whether a number is odd, see whether it's one more than some even number: 7 is odd since it's one more than 6, which is even.
Another way to say this is that zero is even since it can be written in the form 2*n, where n is an integer. Odd numbers can be written in the form 2*n + 1. Again, this lets us talk about whether negative numbers are even and odd: -9 is odd since it's one more than -10, which is even.
Every positive integer can be factored into the product of prime numbers, and there's only one way to do it for every number. For instance, 280 = 2x2x2x5x7, and there's only one way to factor 280 into prime numbers. This is an important theorem: the Fundamental Theorem of Arithmetic. See Notes and Literature on Prime Numbers from Understanding Mathematics by Peter Alfeld, and the Largest Known Primes page.
Most mathematicians, at least when they're talking to each other, use Z to refer to the set of integers. In German the word "zahlen" means "to count" and "Zahl" means "number." Mathematicians also use the letter N to talk about the set of positive integers, in other words the set {1,2,3,4,5,6, ...}.
From the Dr. Math archives:
Adding and Subtracting Integers (Elementary/Addition)
Consecutive Integers (High School/Algebra)
Introduction to Negative Numbers (Elementary/Subtraction)
Is Zero Even, Odd, or Neither? (Elementary/About Numbers)
Sets and Integer Pairs (High School/Discrete Math)
Sets and Subsets (Middle School/Algebra)
Partitioning the Integers (High School/Discrete Math)
Why is 1 not considered prime? (Middle School/About Numbers)
On the Web
Integers
6th Grade Math Class Integer Pictures - Oak Point Intermediate School
--------------------------------------------------------------------------------
Rational Numbers 5/1, 1/2, 1.75, -97/3 ...
A rational number is any number that can be written as a ratio of two integers (hence the name!). In other words, a number is rational if we can write it as a fraction where the numerator and denominator are both integers.
The term "rational" comes from the word "ratio," because the rational numbers are the ones that can be written in the ratio form p/q where p and q are integers. Irrational, then, just means all the numbers that aren't rational.
Every integer is a rational number, since each integer n can be written in the form n/1. For example 5 = 5/1 and thus 5 is a rational number. However, numbers like 1/2, 45454737/2424242, and -3/7 are also rational, since they are fractions whose numerator and denominator are integers.
So the set of all rational numbers will contain the numbers 4/5, -8, 1.75 (which is 7/4), -97/3, and so on.
Is .999 repeating a rational number? Well, a number is rational if it can be written as A/B (A over B): .3 = 3/10 and .55555..... = 5/9, so these are both rational numbers. Now look at .99999999..... which is equal to 9/9 = 1. We have just written down 1 and .9999999 in the form A/B where A and B are both 9, so 1 and .9999999 are both rational numbers. In fact all repeating decimals like .575757575757... , all integers like 46, and all finite decimals like .472 are rational.
From the Dr. Math Archives:
Rational and Irrational Numbers (Elementary/About Numbers)
About Rational Numbers (Middle School/About Numbers)
Is a Ratio Rational or Irrational? (High School/Analysis)
On the Web:
Concrete Algebra: Numbers
Rational Number - Eric Weisstein's World of Mathematics
Rational Numbers
Real Numbers
Complex Numbers
--------------------------------------------------------------------------------
Irrational Numbers sqrt(2), pi, e, the Golden Ratio ...
Irrational numbers are numbers that can be written as decimals but not as fractions.
An irrational number is any real number that is not rational. By real number we mean, loosely, a number that we can conceive of in this world, one with no square roots of negative numbers (such a number is called complex.)
A real number is a number that is somewhere on a number line, so any number on a number line that isn't a rational number is irrational. The square root of 2 is an irrational number because it can't be written as a ratio of two integers.
Other irrational numbers include the square root of 3, the square root of 5, pi, e, and the golden ratio. (For more information about pi and e, see Pi = 3.14159... and E = 2.71828..., also from the Dr. Math FAQ.)
Pi is an irrational number because it cannot be expressed as a ratio (fraction) of two integers: it has no exact decimal equivalent, although 3.1415926 is good enough for many applications. The square root of 2 is another irrational number that cannot be written as a fraction.
In mathematics, a name can be used with a very precise meaning that may have little to do with the meaning of the English word. ("Irrational" numbers are NOT numbers that can't argue logically!)
From the Dr. Math Archives:
Venn Diagram of Our Number System (Middle School/About Numbers)
Golden Ratio and Golden Rectangle (Elementary/Golden Rectangle)
Irrational Pi (High School/Transcendental Numbers)
The Number e (High School/Transcendental Numbers)
Meaning of Irrational Exponents (High School/Algebra)
Irrational Powers (College/Modern Algebra)
Proof that Sqrt(2) is Irrational (High School/Square Roots)
Proving the Square Root of 3 Irrational (High School/Square Roots)
Are Transcendentals Irrational? (High School/Transcendental Numbers)
On the Web:
MATH 140 In Class Exercise on E-Primes Name____________________
Let E be the set of all positive even numbers: E = {2,4,6,8,10,…}.
We will consider only those numbers in this exercise.
1. Factor each of the following numbers in E – that is – if the number can be factored into a product of numbers in E write down the factorization. Break it into as many factors as possible. No odd numbers are allowed. If it can’t be factored just write the number.
n
2
4
6
8
10
12
14
16
factor
n
18
20
22
24
26
28
30
32
factor
2. An E-Prime is a number in E that can not be factored. 2 for example is an E-Prime since there is no factorization of 2 except 2 = 2.
List the E-Primes from 2 – 32.
______________________________________________________________
What do the E-Primes have in common? _______________
3. Is 50 an E-Prime? Why or why not? _______________
4. Factor 60 as a product of E-Primes in two different ways, that is with different prime factors.
5. Find another even number that can be factored into a product of E-Primes in two different ways.
www.wou.edu/~beaverc/211Fall2005/Chapter 4 Connect.pdf
glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4
The Database of Integer Sequences, Part 11
Part of the On-Line Encyclopedia of Integer Sequences
This is a section of the main database for the On-Line Encyclopedia of Integer Sequences.
For more information see the following pages:
( www.research.att.com/~njas/sequences/ then )
Seis.html: Welcome
index.html: Lookup
indexfr.html: Francais
demo1.html: Demos
Sindx.html: Index
WebCam.html: WebCam
Submit.html: Contribute new sequence or comment
eishelp1.html: Internal format
eishelp2.html: Beautified format
transforms.html: Transforms
Spuzzle.html: Puzzles
Shot.html: Hot
classic.html: Classics
ol.html: Superseeker
JIS/index.html: Journal of Integer Sequences
pages.html: More pages
Maintained by: N. J. A. Sloane (njas@research.att.com),
home page: www.research.att.com/~njas/
(start)
%I A068323
%S A068323 0,1,1,1,2,2,2,2,2,3,1,3,2,3,4,3,1,4,2,4,4,4,1,5,2,4,1,4,1,6,2,3,5,5,2,
%T A068323 6,1,3,5,5,1,7,2,5,3,5,1,7,2,6,5,5,1,7
%N A068323 a(n)=number of arithmetic progressions of primes, nondecreasing with sum n.
%Y A068323 Sequence in context: A100825 A008767 A105255 this_sequence A054990 A046921 A078178
%Y A068323 Adjacent sequences: A068320 A068321 A068322 this_sequence A068324 A068325 A068326
%K A068323 easy,nonn
%O A068323 1,5
%A A068323 Naohiro Nomoto (n_nomoto(AT)yabumi.com), Feb 27 2002
%I A054990
%S A054990 1,1,1,2,2,2,2,2,3,2,1,3,2,2,3,5,3,6,2,2,3,3,4,2,2,2,1,2,3,5,4,4,5,2,5,
%T A054990 6,1,2,4,7,1,3,4,3,3,3,4,2,5,5,6,4,4,2,2,4,3,4,2,4,4,3,5,3,4,5,4,5,6,5,
%U A054990 2,7,1,4,2,3,1,6,3,4,7,3,3,3,5,5,4,3,8,3,6,2,4,3,4,5,6,6,5,5,4,5
%N A054990 Number of prime divisors of n! + 1 (counted with multiplicity).
%C A054990 The smallest k! with n prime factors occurs for n in A060250.
%C A054990 103!+1 = 27437*31084943*C153, so a(103) is unknown until this 153-digit composite is factored. a(104) = 4 and a(105) = 6. - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jun 10 2003
%H A054990 Hisanori Mishima,
Factorizations of many number sequences
%H A054990 Hisanori Mishima,
Factorizations of many number sequences
%H A054990 R. G. Wilson v,
Explicit factorizations
%H A054990 Paul Leyland,
Factors of n!+1.
%e A054990 a(2)=2 because 4! + 1 = 25 = 5*5
%t A054990 a[q_] := Module[{x, n}, x=FactorInteger[q!+1]; n=Length[x]; Sum[Table[x[[i]][[2]], {i, n}][[j]], {j, n}]]
%o A054990 (PARI) for(n=1,64,print1(bigomega(n!+1),","))
%Y A054990 Cf. A000040 (prime numbers), A001359 (twin primes). Also A054988, A054989, A054991, A054992.
%Y A054990 Cf. A066856 (number of distinct prime divisors of n!+1), A084846 (mu(n!+1)).
%Y A054990 Sequence in context: A008767 A105255 A068323 this_sequence A046921 A078178 A105068
%Y A054990 Adjacent sequences: A054987 A054988 A054989 this_sequence A054991 A054992 A054993
%K A054990 nonn,hard
%O A054990 1,4
%A A054990 Arne Ring (arne.ring(AT)epost.de), May 30 2000
%E A054990 More terms from Robert G. Wilson V (rgwv(AT)rgwv.com), Mar 23 2001
%E A054990 More terms from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jun 10 2003
%I A046921
%S A046921 1,2,2,2,2,2,3,2,1,4,3,2,3,1,2,4,2,2,4,3,2,3,3,2,4,3,2,5,1,2,6,3,1,
%T A046921 3,4,2,5,4,2,6,3,2,4,2,3,6,2,1,4,3,4,6,4,2,6,5,2,6,3,2,5,1,2,3,5,4,
%U A046921 5,4,1,8,4,1,6,3,2,6,2,2,6,6,1,4,5,3,7,4,3,6,2,3,10,2,3,4,4,3,3,4,2
%N A046921 Number of ways to express 2n+1 as p+2a^2; p = 1 or prime, a >= 0.
%C A046921 Goldbach conjectured this sequence is never zero.
%H A046921 L. Hodges,
A lesser-known Goldbach conjecture, Math. Mag., 66 (1993), 45-47.
%H A046921
Index entries for sequences related to Goldbach conjecture
%Y A046921 Sequence in context: A105255 A068323 A054990 this_sequence A078178 A105068 A120676
%Y A046921 Adjacent sequences: A046918 A046919 A046920 this_sequence A046922 A046923 A046924
%K A046921 nonn
%O A046921 0,2
%A A046921 David W. Wilson (davidwwilson(AT)comcast.net)
%I A078178
%S A078178 2,2,2,2,2,3,2,2,2,2,4,2,16,2,2,4,3,2,2,2,7,4,2,3,2,3,2,10,2,2,108,3,6,
%T A078178 2,3,7,2,2,4,2,16,3,2,2,2,20,2,7,2,3,3,2,2,2,2,9,4,2,2,7,8,3,2,2,2,24,
%U A078178 2,6,2,12,4,3,8,6,2,4,3,9,194,3,13,2,8,2,2,3,8,2,10,6,4,2,2,54,2,132,4
%N A078178 Least k>=2 such that n^k + n - 1 is prime.
%C A078178 n^a(n) + n - 1 = A078179(n).
%e A078178 7^2+7-1=5*11, but 7^3+7-1=349=A000040(70), therefore a(7)=3.
%Y A078178 Cf. A076845, A078179.
%Y A078178 Sequence in context: A068323 A054990 A046921 this_sequence A105068 A120676 A001031
%Y A078178 Adjacent sequences: A078175 A078176 A078177 this_sequence A078179 A078180 A078181
%K A078178 nonn
%O A078178 2,1
%A A078178 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Nov 20 2002
%E A078178 More terms from Benoit Cloitre, Nov 20 2002
%I A105068
%S A105068 1,1,1,2,2,2,2,2,3,2,2,2,3,2,4,4,1,4,5,4,2,2,4,2,2,3,4,3,4,3,6,4,3,2,4,
%T A105068 4,5,3,5,5,4,5,7,4,5,4,7,5,4,4
%N A105068 Number of distinct prime divisors of 10^n - 3.
%e A105068 If n=1, 2 or 3, then 10^n - 3 = prime and thus the first three terms are 1.
%t A105068 Table[Length[FactorInteger[10^n - 3]], {n, 1, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Feb 21 2006
%Y A105068 Sequence in context: A054990 A046921 A078178 this_sequence A120676 A001031 A035250
%Y A105068 Adjacent sequences: A105065 A105066 A105067 this_sequence A105069 A105070 A105071
%K A105068 nonn
%O A105068 1,4
%A A105068 Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Apr 05 2005
%E A105068 More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Feb 21 2006
%I A120676
%S A120676 1,2,2,2,2,2,3,2,2,3,2,2,2,3,3,2,3,2,2,2,3,2,3,3,2,2,3,2,3,2,2,3,2,2,3,
%T A120676 3,2,3,3,3,2,2,2,4,2,2,3,2,3,3,3,2,3,2,3,2,2,3,3,3,2,2,3,2,3,3,2,4,2,2,
%U A120676 3,2,2,3,3,3,2,2,4,2,2,3,3,3,3,2,3,3,3,3,3,2,2,2,4,2,3,3,2,2,3,3,2,3,4
%N A120676 Number of prime factors of even square-free numbers A039956.
%F A120676 a(n)=A001221(A039956(n))=A001222(A039956(n))=A120675(n)+1.
%p A120676 issquarefree := proc(n::integer) local nf, ifa ; nf := op(2,ifactors(n)) ; for ifa from 1 to nops(nf) do if op(2,op(ifa,nf)) >= 2 then RETURN(false) ; fi ; od : RETURN(true) ; end: A001221 := proc(n::integer) RETURN(nops(numtheory[factorset](n))) ; end: A039956 := proc(maxn) local n,a ; a := [2] ; for n from 4 to maxn by 2 do if issquarefree(n) then a := [op(a),n] ; fi ; od : RETURN(a) ; end: A120676 := proc(maxn) local a,n; a := A039956(maxn) ; for n from 1 to nops(a) do a := subsop(n=A001221(a[n]),a) ; od ; RETURN(a) ; end: nmax := 600 : a := A120676(nmax) : for n from 1 to nops(a) do printf("%d,",a[n]) ; od ; - Richard J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 17 2006
%Y A120676 Sequence in context: A046921 A078178 A105068 this_sequence A001031 A035250 A067743
%Y A120676 Adjacent sequences: A120673 A120674 A120675 this_sequence A120677 A120678 A120679
%K A120676 nonn,new
%O A120676 1,2
%A A120676 Lekraj Beedassy (boodhiman(AT)yahoo.com), Jun 24 2006
%E A120676 Corrected and extended by Richard J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 17 2006
%I A001031 M0213 N0077
%S A001031 1,2,2,2,2,2,3,2,3,3,3,4,3,2,4,3,4,4,3,3,5,4,4,6,4,3,6,3,4,7,4,5,6,3,5,
%T A001031 7,6,5,7,5,5,9,5,4,10,4,5,7,4,6,9,6,6,9,7,7,11,6,6,12,4,5,10,4,7,10,6,5,
%U A001031 9,8,8,11,6,5,13,5,8,11,6,8,10,6,6,14,9,6,12,7,7,15,7,8,13,5,8,12,8,9
%N A001031 a(n) = number of decompositions of 2n into sum of two primes (counting 1 as a prime).
%D A001031 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 9.
%D A001031 Richard K. Guy, Unsolved problems in number theory, second edition, Springer-Verlag, 1994.
%D A001031 G. H. Hardy and J. E. Littlewood, Some problems of `partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1922.
%D A001031 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 79.
%D A001031 J. Richstein, Verifying the Goldbach conjecture up to 4*10^14, Mathematics of Computation, Vol. 70, No. 236, pp. 1745-1749, July 2000.
%D A001031 Matti K. Sinisalo, Checking the Goldbach conjecture up to 4*10^11, Mathematics of Computation, Vol. 61, No. 204, pp. 931-934, October 1993.
%D A001031 M. L. Stein and P. R. Stein, Tables of the Number of Binary Decompositions of All Even Numbers Less Than 200,000 into Prime Numbers and Lucky Numbers. Report LA-3106, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Sep 1964.
%D A001031 Apostolos Doxiadis: Uncle Petros and Goldbach's Conjecture, Faber and Faber, 2001
%H A001031 T. Oliveira e Silva,
Goldbach conjecture verification
%H A001031 Eric Weisstein's World of Mathematics,
Goldbach Partition
%H A001031
Index entries for sequences related to Goldbach conjecture
%Y A001031 Cf. A002372, A002373, A002374, A002375, A045917, A006307.
%Y A001031 Sequence in context: A078178 A105068 A120676 this_sequence A035250 A067743 A029230
%Y A001031 Adjacent sequences: A001028 A001029 A001030 this_sequence A001032 A001033 A001034
%K A001031 nonn,easy,nice
%O A001031 1,2
%A A001031 njas
%E A001031 More terms from Ray Chandler (RayChandler(AT)alumni.tcu.edu), Sep 19 2003
%I A035250
%S A035250 1,2,2,2,2,2,3,2,3,4,4,4,4,3,4,5,5,4,5,4,5,6,6,6,6,6,7,7,7,7,8,7,7,8,8,
%T A035250 9,10,9,9,10,10,10,10,9,10,10,10,9,10,10,11,12,12,12,13,13,14,14,14,13,
%U A035250 13,12,12,13,13,14,14,13,14,15,15,14,14,13,14,15,15,15,16,15,15,16,16
%N A035250 Number of primes between n and 2n (inclusive).
%C A035250 By Bertrand's Postulate (proved by Chebyshev), there is always a prime between n and 2n, i.e. a(n) is positive for all n.
%D A035250 Aigner, M. and Ziegler, G. Proofs from The Book (2nd edition). Springer-Verlag, 2001.
%H A035250 International Mathematics Olympiad,
Proof of Bertrand's Postulate
%e A035250 The primes between n = 13 and 2n = 26, inclusive, are 13, 17, 19, 23; so a(13) = 4.
%Y A035250 Sequence in context: A105068 A120676 A001031 this_sequence A067743 A029230 A084294
%Y A035250 Adjacent sequences: A035247 A035248 A035249 this_sequence A035251 A035252 A035253
%K A035250 nonn
%O A035250 1,2
%A A035250 Erich Friedman (erich.friedman(AT)stetson.edu)
%I A067743
%S A067743 0,1,2,2,2,2,2,3,2,4,2,4,2,4,2,4,2,5,2,4,4,4,2,6,2,4,4,4,2,6,2,5,4,4,2,
%T A067743 8,2,4,4,6,2,6,2,6,4,4,2,8,2,5,4,6,2,6,4,6,4,4,2,10,2,4,4,6,4,6,2,6,4,
%U A067743 6,2,9,2,4,6,6,2,8,2,8,4,4,2,10,4,4,4,6,2,10,2,6,4,4,4,10,2,5,4,8,2,8
%N A067743 Number of divisors of n not in the half-open interval [sqrt(n/2), sqrt(n*2)).
%D A067743 Problem 10847, Amer. Math. Monthly 109, (2002), p. 80.
%F A067743 A000005[n] - A067742[n]
%e A067743 a(6)=2 because 2 divisors of 6 (ie 1 and 6) fall outside sqrt(3) to sqrt(12).
%Y A067743 Cf. A067742, A000005.
%Y A067743 Sequence in context: A120676 A001031 A035250 this_sequence A029230 A084294 A067752
%Y A067743 Adjacent sequences: A067740 A067741 A067742 this_sequence A067744 A067745 A067746
%K A067743 easy,nonn
%O A067743 1,3
%A A067743 Marc LeBrun (mlb(AT)well.com), Jan 29 2002
%I A029230
%S A029230 1,0,1,0,1,0,1,1,1,2,2,2,2,2,3,2,4,3,5,4,6,5,6,6,7,7,8,
%T A029230 9,10,10,12,11,13,12,15,14,17,17,19,19,21,21,23,23,26,26,
%U A029230 29,29,32,32,35,35,38,38,42,42,46,46,50,50,54,54,58,59
%N A029230 Expansion of 1/((1-x^2)(1-x^7)(1-x^9)(1-x^10)).
%Y A029230 Sequence in context: A001031 A035250 A067743 this_sequence A084294 A067752 A025422
%Y A029230 Adjacent sequences: A029227 A029228 A029229 this_sequence A029231 A029232 A029233
%K A029230 nonn
%O A029230 0,10
%A A029230 njas
%I A084294
%S A084294 2,2,2,2,2,3,3,2,3,3,3,4,4,3,4,4,5,5,5,5,4,5,5,7,6,6,5,5,5,5,7,7,7,7,8,
%T A084294 7,8,9,8,8,7,7,9,8,9,8,9,11,10,10,11,10,10,9,10,11,10,9,9,9,8,10,11,11,
%U A084294 10,11,12,12,12,12,12,12,13,13,13,13,14,14,14,14,15,15,15,14,15,14,13
%N A084294 Number of primes in [p(n),n+p(n)] closed interval, where p(n) is the n-th prime.
%F A084294 a(n)=Pi[n+p(n)]=A000720(n+A000040(n))
%t A084294 t[x_] := Table[w, {w, Prime[x], x+Prime[x]}] Table[Count[PrimeQ[t[n]], True], {n, 1, 128}] or Table[PrimePi[n+Prime[n]]-n+1, {n, 1, 25}];
%Y A084294 Cf. A000040, A000720, A061067, A061068, A084295.
%Y A084294 Sequence in context: A035250 A067743 A029230 this_sequence A067752 A025422 A078640
%Y A084294 Adjacent sequences: A084291 A084292 A084293 this_sequence A084295 A084296 A084297
%K A084294 nonn
%O A084294 1,1
%A A084294 Labos E. (labos(AT)ana.sote.hu), May 27 2003
%I A067752
%S A067752 1,1,2,2,2,2,2,3,3,2,3,4,2,3,4,4,3,3,3,5,4,2,4,6,3,4,5,4,4,4,4,6,4,3,6,
%T A067752 7,2,4,6,6,5,4,3,7,6,3,6,8,4,5,6,5,4,6,6,9,4,2,7,8,4,5,8,7,6,6,3,8,6,4,
%U A067752 8,9,3,6,8,7,6,4,6,11,7,3,7,10,4,6,8,6,7
%N A067752 Number of unordered solutions of xy+xz+yz=n in nonnegative integers.
%C A067752 An upper bound on the number of solutions appears to be 1.5*sqrt(n). - T. D. Noe (noe(AT)sspectra.com), Jun 14 2006
%H A067752 T. D. Noe,
Table of n, a(n) for n = 1..10000
%e A067752 a(12)=4 because of (0,1,12),(0,2,6),(0,3,4),(2,2,2).
%e A067752 a(20)=5 because of (0,1,20),(0,2,10),(0,4,5),(1,2,6),(2,2,4).
%t A067752 Table[cnt=0; Do[z=(n-x*y)/(x+y); If[IntegerQ[z], cnt++ ], {x,0,Sqrt[n/3]}, {y,Max[1,x],Sqrt[x^2+n]-x}]; cnt, {n,100}] - T. D. Noe (noe(AT)sspectra.com), Jun 14 2006
%Y A067752 Cf. A067751, A067753, A067754.
%Y A067752 Sequence in context: A067743 A029230 A084294 this_sequence A025422 A078640 A006374
%Y A067752 Adjacent sequences: A067749 A067750 A067751 this_sequence A067753 A067754 A067755
%K A067752 easy,nonn
%O A067752 1,3
%A A067752 Colin L. Mallows (colinm(AT)avaya.com), Jan 31 2002
%E A067752 Corrected, extended, and edited by John W. Layman (layman(AT)math.vt.edu, Dec 3 2004
%I A025422
%S A025422 1,1,1,1,2,2,2,2,2,3,3,2,3,4,3,3,4,4,5,4,5,6,6,4,5,7,6,6,7,8,8,7,6,8,9,7,
%T A025422 10,11,10,9,10,10,11,10,10,15,12,10,11,13,14,12,15,16,18,15,13,16,17,14,
%U A025422 16,20,17,18,16,18,21,18,19,23,24,18,21,21,22,22,23,26,25,24,21,27,27,23
%N A025422 Number of partitions of n into 7 squares.
%Y A025422 Sequence in context: A029230 A084294 A067752 this_sequence A078640 A006374 A064876
%Y A025422 Adjacent sequences: A025419 A025420 A025421 this_sequence A025423 A025424 A025425
%K A025422 nonn
%O A025422 0,5
%A A025422 David W. Wilson (davidwwilson(AT)comcast.net)
%I A078640
%S A078640 1,1,1,1,1,2,2,2,2,2,3,3,2,3,4,5,5,5,4,4,5,6,6,6,6,7,8,6,7,8,9,9,6,7,8,
%T A078640 11,10,8,9,9,11,11,10,10,11,14,13,12,11,12
%N A078640 Number of numbers between 1 and n-1 that are coprime to n(n+1)(n+2).
%C A078640 Does every integer appear?
%e A078640 a(10) is the number of numbers between 1 and 9 that are coprime to 10.11.12, which leaves 1 and 7, hence a(10)=2.
%o A078640 (PARI) newphi(v)=local(vl,fl,np); vl=length(v); np=0; for (s=1,v[1],fl=false; for (r=1,vl,if (gcd(s,v[r])>1,fl=true; break)); if (fl==false,np++)); np v=vector(3); for (i=1,50,v[1]=i; v[2]=i+1; v[3]=i+2; print1(newphi(v)","))
%Y A078640 Sequence in context: A084294 A067752 A025422 this_sequence A006374 A064876 A105517
%Y A078640 Adjacent sequences: A078637 A078638 A078639 this_sequence A078641 A078642 A078643
%K A078640 nonn
%O A078640 1,6
%A A078640 Jon Perry (perry(AT)globalnet.co.uk), Dec 12 2002
%I A006374 M0214
%S A006374 1,1,2,2,2,2,2,3,3,2,4,4,2,4,4,4,4,3,4,6,4,2,6,6,3,6,6,4,6,4,6,7,4,
%T A006374 4,8,8,2,6,8,6,8,4,4,10,6,4,10,8,5,7,8,6,6,8,8,12,4,2,12,8,6,8,10,8,
%U A006374 8,8,4,12,8,4,14,9,4,10,10,10,8,4,10,14,9,4,12,12,4,10,12,6,12,10,8
%N A006374 Number of reduced binary quadratic forms of determinant n.
%D A006374 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 360.
%D A006374 A. Hurwitz and N. Kritikos, transl. with add. material by W . C. Schulz, "Lectures on Number Theory", Springer-Verlag, New York, 1986, p.186.
%Y A006374 Cf. A006371, A006375, A096446, A096445.
%Y A006374 Sequence in context: A067752 A025422 A078640 this_sequence A064876 A105517 A056813
%Y A006374 Adjacent sequences: A006371 A006372 A006373 this_sequence A006375 A006376 A006377
%K A006374 nonn,nice,easy
%O A006374 1,3
%A A006374 njas
%I A064876
%S A064876 0,1,1,1,2,2,2,2,2,3,3,3,2,3,3,3,4,4,3,3,4,4,3,3,4,5,5,5,5,5,5,5,4,4,5,
%T A064876 5,6,6,6,6,6,5,5,5,6,6,6,6,4,7,7,7,6,7,7,7,6,7,7,7,7,6,6,6,8,8,8,7,8,8,
%U A064876 6,6,6,8,7,7,6,8,7,7,8,9,9,9,8,9,9,9,6,8,9,9,9,8,9,9,8,9,7,7,10,10,10
%N A064876 Last of four sequences representing the lexicographical minimal decomposition of n in 4 squares: n = A064873(n)^2 + A064874(n)^2 + A064875(n)^2 + a(n)^2.
%e A064876 a(18) = 3: 18 = A064873(18)^2 + A064874(18)^2 + A064875(18)^2 + a(18)^2 = 0 + 0 + 9 + 9 and the other decompositions (0, 1, 1, 4) and (1, 2, 2, 3) are greater than (0, 0, 3, 3).
%Y A064876 Cf. A064873, A064874, A064875, A064877.
%Y A064876 Sequence in context: A025422 A078640 A006374 this_sequence A105517 A056813 A077430
%Y A064876 Adjacent sequences: A064873 A064874 A064875 this_sequence A064877 A064878 A064879
%K A064876 nonn
%O A064876 0,5
%A A064876 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Oct 10 2001
%I A105517
%S A105517 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,
%T A105517 1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,
%U A105517 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5
%N A105517 Number of times 7 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.
%C A105517 a(n) = #{k: A008963(k) = 7 and 0<=k<=n};
%C A105517 a(A105507(n)) = a(A105507(n) - 1) + 1;
%C A105517 n = A105511(n) + A105512(n) + A105513(n) + A105514(n) + A105515(n) + A105516(n) + a(n) + A105518(n) + A105519(n).
%Y A105517 Cf. A000030, A000045.
%Y A105517 Sequence in context: A078640 A006374 A064876 this_sequence A056813 A077430 A105513
%Y A105517 Adjacent sequences: A105514 A105515 A105516 this_sequence A105518 A105519 A105520
%K A105517 nonn,base
%O A105517 0,45
%A A105517 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 11 2005
%I A056813
%S A056813 1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,
%T A056813 5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,
%U A056813 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7
%N A056813 Largest non-unitary prime factor of LCM[1,...,n]; i.e. the largest prime which occurs to power > 1 in prime factorization of LCM[1,..,n].
%C A056813 For n>0, p(n) appears {(p(n+1))^2 - (p(n))^2} times [from n=(p(n))^2 to n=(p(n+1))^2 - 1], i.e., A000040(n) appears A069482(n) times[from n=A001248(n) to n=A084920(n+1)] - Lekraj Beedassy (boodhiman(AT)yahoo.com), Mar 31 2005
%F A056813 a(n)=p(w) if p(w)^2 <= n < p(w+1)^2.
%e A056813 The j-th prime appears at the position of its square, at n=P(j)^2.
%Y A056813 A056168, A056170.
%Y A056813 Sequence in context: A006374 A064876 A105517 this_sequence A077430 A105513 A004233
%Y A056813 Adjacent sequences: A056810 A056811 A056812 this_sequence A056814 A056815 A056816
%K A056813 nonn
%O A056813 0,4
%A A056813 Labos E. (labos(AT)ana.sote.hu), Aug 28 2000
%I A077430
%S A077430 1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,
%T A077430 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%U A077430 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5
%N A077430 Floor(Log10(2*n^2)) + 1.
%Y A077430 Cf. A077431, A077432, A004216, A077429, A077433.
%Y A077430 Sequence in context: A064876 A105517 A056813 this_sequence A105513 A004233 A068549
%Y A077430 Adjacent sequences: A077427 A077428 A077429 this_sequence A077431 A077432 A077433
%K A077430 nonn
%O A077430 1,3
%A A077430 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Nov 05 2002
%I A105513
%S A105513 0,0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,
%T A105513 5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,
%U A105513 8,9,9,9,9,9,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11
%N A105513 Number of times 3 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.
%C A105513 a(n) = #{k: A008963(k) = 3 and 0<=k<=n};
%C A105513 a(A105503(n)) = a(A105503(n) - 1) + 1;
%C A105513 n = A105511(n) + A105512(n) + a(n) + A105514(n) + A105515(n) + A105516(n) + A105517(n) + A105518(n) + A105519(n).
%Y A105513 Cf. A000030, A000045.
%Y A105513 Sequence in context: A105517 A056813 A077430 this_sequence A004233 A068549 A023968
%Y A105513 Adjacent sequences: A105510 A105511 A105512 this_sequence A105514 A105515 A105516
%K A105513 nonn,base
%O A105513 0,10
%A A105513 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 11 2005
%I A004233
%S A004233 0,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,
%T A004233 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%U A004233 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5
%N A004233 ln(n) rounded up.
%Y A004233 Sequence in context: A056813 A077430 A105513 this_sequence A068549 A023968 A000196
%Y A004233 Adjacent sequences: A004230 A004231 A004232 this_sequence A004234 A004235 A004236
%K A004233 nonn
%O A004233 1,3
%A A004233 njas
%I A068549
%S A068549 2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,
%T A068549 5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,
%U A068549 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,11,11,11,11,11,11,11,11,11,11,11
%N A068549 Largest prime <= sqrt(2n-4) - 1.
%D A068549 F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 19.
%Y A068549 Sequence in context: A077430 A105513 A004233 this_sequence A023968 A000196 A111850
%Y A068549 Adjacent sequences: A068546 A068547 A068548 this_sequence A068550 A068551 A068552
%K A068549 nonn
%O A068549 10,1
%A A068549 njas, Mar 22 2002
%I A023968
%S A023968 0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%T A023968 4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,
%U A023968 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6
%N A023968 First digit after decimal point of 9-th root of n.
%t A023968 Array[ Function[ n, RealDigits[ N[ Power[ n, 1/9 ], 10 ], 10 ]// (#[ [ 1, #[ [ 2 ] ]+1 ] ])& ], 110 ]
%Y A023968 Sequence in context: A105513 A004233 A068549 this_sequence A000196 A111850 A059396
%Y A023968 Adjacent sequences: A023965 A023966 A023967 this_sequence A023969 A023970 A023971
%K A023968 nonn,base
%O A023968 1,6
%A A023968 njas, Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A000196
%S A000196 0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,
%T A000196 5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,
%U A000196 8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10
%N A000196 Integer part of square root of n. Or, number of squares <= n. Or, n appears 2n+1 times.
%C A000196 Also the integer part of the geometric mean of the divisors of n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Dec 19 2001
%C A000196 a(n)=Card(k, 0
%C A000196 Number of numbers k (<=n) with an odd number of divisors - Benoit Cloitre (abmt(AT)wanadoo.fr), Sep 07 2002
%D A000196 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 23.
%D A000196 K. Atanassov, On the 100-th, 101-st and the 102-th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 3, 94-96.
%D A000196 K. Atanassov, On Some of Smarandache's Problems, American Research Press, 1999, 16-21.
%D A000196 N. J. A. Sloane and A. R. Wilks, On sequences of Recaman type, paper in preparation, 2006.
%D A000196 F. Smarandache, Only Problems, not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.
%H A000196 K. Atanassov,
On Some of the Smarandache's Problems
%H A000196 H. Bottomley,
Illustration of A000196, A048760, A053186
%H A000196 M. L. Perez et al., eds.,
Smarandache Notions Journal
%H A000196 F. Smarandache,
Only Problems, Not Solutions!.
%F A000196 a(n) = a(n-1) + floor(n/(a(n-1)+1)^2), a(0) = 0. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 12 2004
%p A000196 Digits := 100; A000196 := n->floor(evalf(sqrt(n)));
%o A000196 (PARI) a(n)=floor(sqrt(n))
%o A000196 (PARI) a(n)=sqrtint(n)
%Y A000196 [A000267(n)/2]=A000196(n). Cf. A028391, A048766, A003056.
%Y A000196 Cf. A079051.
%Y A000196 Sequence in context: A004233 A068549 A023968 this_sequence A111850 A059396 A108602
%Y A000196 Adjacent sequences: A000193 A000194 A000195 this_sequence A000197 A000198 A000199
%K A000196 nonn,easy,nice
%O A000196 0,5
%A A000196 njas
%I A111850
%S A111850 1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,6,
%T A111850 7,8,8,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,
%U A111850 11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,14,14,14,14,14
%N A111850 Number of numbers m <= n such that 0 equals the first digit after decimal point of square root of n in decimal representation.
%C A111850 For n>1: if A023961(n)=0 then a(n)=a(n-1)+1 else a(n)=a(n-1).
%C A111850 a(n)/n --> 1/10.
%D A111850 G. Polya and G. Szego, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Two, Chap. 4, Sect. 4, Problem 178.
%e A111850 a(10) = 3, a(100) = 15, a(1000) = 118, a(10000) = 1050.
%Y A111850 Cf. A111851, A111852, A111853, A111854, A111855, A111856, A111857, A111858, A111859, A111890.
%Y A111850 Sequence in context: A068549 A023968 A000196 this_sequence A059396 A108602 A085290
%Y A111850 Adjacent sequences: A111847 A111848 A111849 this_sequence A111851 A111852 A111853
%K A111850 nonn,base
%O A111850 1,4
%A A111850 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 20 2005
%I A059396
%S A059396 0,0,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,
%T A059396 5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,
%U A059396 7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9
%N A059396 Number of primes less than square root of n-th prime; i.e. number of trial divisions by smaller primes to show that n-th prime is indeed prime.
%C A059396 Perhaps close to 2*(n/loge(n))^(1/2)
%F A059396 a(n) = A000720(A000196(A000040(n)))
%e A059396 a(32) = 5 since the 32nd prime is 131 which is not divisible by 2, 3, 5, 7 or 11 (and does not need to be tested against 13, 17, 19 etc. since 13^2 = 169>131).
%Y A059396 Sequence in context: A023968 A000196 A111850 this_sequence A108602 A085290 A108611
%Y A059396 Adjacent sequences: A059393 A059394 A059395 this_sequence A059397 A059398 A059399
%K A059396 nonn
%O A059396 0,5
%A A059396 Henry Bottomley (se16(AT)btinternet.com), Jan 29 2001
%I A108602
%S A108602 0,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,4,4,5,5,5,5,5,5,5,5,
%T A108602 5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,6,7,7,7,7,7,7,7,7,7,7,8,7,7,8,8,7,
%U A108602 8,8,8,8,8,8,8,8,8,8,8,8,9,8,9,8,9,8,9,9,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9
%N A108602 Number of distinct prime factors of highly composite numbers (definition 1, A002182).
%F A108602 a(n) = A001221(A002182(n)).
%e A108602 A002182(8) = 48 = 2^4*3, which has 2 distinct prime factors, so a(8)=2.
%Y A108602 Cf. A002182, A002183.
%Y A108602 Sequence in context: A000196 A111850 A059396 this_sequence A085290 A108611 A104355
%Y A108602 Adjacent sequences: A108599 A108600 A108601 this_sequence A108603 A108604 A108605
%K A108602 nonn
%O A108602 1,4
%A A108602 Jud McCranie (j.mccranie(AT)adelphia.net), Jun 12 2005
%E A108602 Edited by Ray Chandler (RayChandler(AT)alumni.tcu.edu), Nov 11 2005
%I A085290
%S A085290 2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,8,8,8,
%T A085290 8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,11,11,11,11,11,11,11,11,11,
%U A085290 11,11,11,13,13,13,13,13,13,13,13,13,13,13,13,13,16,16,16,16,16,16,16
%N A085290 Max[p1^b1] over all sorted multiplicative partitions of n! of length n.
%H A085290 Eric Weisstein's World of Mathematics,
Alladi-Grinstead Constant
%e A085290 6! = 2*2*2*2*5*9 = 2*2*3*3*4*5, the smallest terms of which are 2 and 2, so a(6)=Max[2,2]=2.
%o A085290 (PARI) works(n, m) = local(f, s, l, p, x); f = factor(n!); s = 0; l = matsize(f)[1]; for (i = 1, l, p = f[i, 1]; x = 1; while (p^x < m, x++); s += f[i, 2]\x; if (f[i, 2] < x, return(0))); s >= n; a(n) = local(f, m); f = factor(n); m = 2; while (works(n, m), m++); m - 1 (Wasserman)
%Y A085290 Cf. A085288, A085289, A085291.
%Y A085290 Cf. A103332.
%Y A085290 Sequence in context: A111850 A059396 A108602 this_sequence A108611 A104355 A092278
%Y A085290 Adjacent sequences: A085287 A085288 A085289 this_sequence A085291 A085292 A085293
%K A085290 nonn
%O A085290 4,1
%A A085290 Eric W. Weisstein (eric(AT)weisstein.com), Jun 23, 2003
%E A085290 More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Jan 31 2005
%I A108611
%S A108611 0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,
%T A108611 6,6,7,7,7,7,7,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,11,11,11,11,11,12,
%U A108611 12,12,12,12,12,13,13,13,13,13,14,14,14,14,14,15,15,15,15,15,16,16,16
%N A108611 Excess of Beatty-function of 1/sin(1) over n.
%F A108611 a(n) = A108120[n] - n.
%Y A108611 Cf. A023800, A031943, A037918, A039215, A043091, A047253, A108120.
%Y A108611 Sequence in context: A059396 A108602 A085290 this_sequence A104355 A092278 A105512
%Y A108611 Adjacent sequences: A108608 A108609 A108610 this_sequence A108612 A108613 A108614
%K A108611 nonn
%O A108611 0,11
%A A108611 Zak Seidov (zakseidov(AT)yahoo.com), Jun 13 2005
%I A104355
%S A104355 0,0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,
%T A104355 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,
%U A104355 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6
%N A104355 Number of trailing zeros in decimal representation of A104350(n).
%C A104355 a(A104356(n)) = n and a(m) < n for m < A104356(n).
%Y A104355 Cf. A027868.
%Y A104355 Sequence in context: A108602 A085290 A108611 this_sequence A092278 A105512 A002266
%Y A104355 Adjacent sequences: A104352 A104353 A104354 this_sequence A104356 A104357 A104358
%K A104355 nonn,base
%O A104355 1,10
%A A104355 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 06 2005
%I A092278
%S A092278 0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,
%T A092278 6,7,7,7,7,7,7,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,12,
%U A092278 12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,15,15,15,15,15,16,16,16,16
%N A092278 Floor( (3*n+4)/16 ).
%D A092278 J. O'Rourke, Art Gallery Theorems and Algorithms, Oxford Univ. Press, 1987, p. 82.
%Y A092278 Sequence in context: A085290 A108611 A104355 this_sequence A105512 A002266 A075249
%Y A092278 Adjacent sequences: A092275 A092276 A092277 this_sequence A092279 A092280 A092281
%K A092278 nonn
%O A092278 0,11
%A A092278 njas, Feb 18 2004
%I A105512
%S A105512 0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,
%T A105512 6,6,7,7,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,10,10,11,11,11,11,11,12,12,12,
%U A105512 12,12,13,13,13,13,13,13,13,13,13,14,14,14,14,14,15,15,15,15,15,16,16
%N A105512 Number of times 2 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.
%C A105512 a(n) = #{k: A008963(k) = 2 and 0<=k<=n};
%C A105512 a(A105502(n)) = a(A105502(n) - 1) + 1;
%C A105512 n = A105511(n) + a(n) + A105513(n) + A105514(n) + A105515(n) + A105516(n) + A105517(n) + A105518(n) + A105519(n).
%Y A105512 Cf. A000030, A000045.
%Y A105512 Sequence in context: A108611 A104355 A092278 this_sequence A002266 A075249 A008648
%Y A105512 Adjacent sequences: A105509 A105510 A105511 this_sequence A105513 A105514 A105515
%K A105512 nonn,base
%O A105512 0,9
%A A105512 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 11 2005
%I A002266
%S A002266 0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,
%T A002266 6,6,7,7,7,7,7,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,11,11,11,11,11,12,
%U A002266 12,12,12,12,13,13,13,13,13,14,14,14,14,14,15,15,15,15,15,16,16,16
%N A002266 Integers repeated 5 times.
%C A002266 For n>3, number of consecutive "11's" after the (n+3) "1's" in the continued fraction for sqrt(L(n+2)/L(n)) where L(n) is the n-th Lucas number A000002 (see example). E.g. the continued fraction for sqrt(L(11)/L(9)) is [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 11, 58, 2, 4, 1, ....] with 12 consecutive ones followed by floor(11/5)=2 elevens. - Benoit Cloitre (abmt(AT)wanadoo.fr), Jan 08 2006
%F A002266 Floor(n/5), n>=0.
%F A002266 G.f.: x^5/((1-x)(1-x^5)).
%Y A002266 Cf. A008648.
%Y A002266 a(n)=A010766(n,5).
%Y A002266 Cf. A004526, A002264, A002265, A010761, A010762, A110532, A110533.
%Y A002266 Sequence in context: A104355 A092278 A105512 this_sequence A075249 A008648 A105511
%Y A002266 Adjacent sequences: A002263 A002264 A002265 this_sequence A002267 A002268 A002269
%K A002266 nonn,easy
%O A002266 0,11
%A A002266 njas
%I A075249
%S A075249 1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,8,8,8,
%T A075249 8,8,9,9,9,9,9,10,10,10,10,10,11,11,11,11,11,12,12,12,12,12,13,14,13,
%U A075249 13,13,14,14,14,14,14,15,15,15,15,15,16,16,16,16,16,17,17,17,17,17,18
%N A075249 x-value of the solution (x,y,z) to 5/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z and having the largest z-value. The y and z components are in A075250 and A075251.
%C A075249 See A075248 for more details.
%F A075249 Is a(n) = A047252(n-3)-n+4 ? - Ralf Stephan, Feb 24 2004
%t A075249 For[xLst={}; yLst={}; zLst={}; n=3, n<=100, n++, cnt=0; xr=n/5; If[IntegerQ[xr], x=xr+1, x=Ceiling[xr]]; While[yr=1/(5/n-1/x); If[IntegerQ[yr], y=yr+1, y=Ceiling[yr]]; cnt==0&&y>x, While[zr=1/(5/n-1/x-1/y); cnt==0&&zr>y, If[IntegerQ[zr], z=zr; cnt++; AppendTo[xLst, x]; AppendTo[yLst, y]; AppendTo[zLst, z]]; y++ ]; x++ ]]; xLst
%Y A075249 Cf. A075248, A075250, A075251.
%Y A075249 Sequence in context: A092278 A105512 A002266 this_sequence A008648 A105511 A027868
%Y A075249 Adjacent sequences: A075246 A075247 A075248 this_sequence A075250 A075251 A075252
%K A075249 hard,nice,nonn
%O A075249 3,3
%A A075249 T. D. Noe (noe(AT)sspectra.com), Sep 10 2002
%I A008648
%S A008648 1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,7,
%T A008648 7,7,7,7,9,9,9,9,9,11,11,11,11,11,13,13,13,13,13,15,15,
%U A008648 15,15,15,18,18,18,18,18,21,21,21,21,21,24,24,24,24,24
%N A008648 Molien series of 3 X 3 upper triangular matrices over GF( 5 ).
%D A008648 D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 105.
%H A008648 INRIA Algorithms Project,
Encyclopedia of Combinatorial Structures 221
%H A008648
Index entries for Molien series
%F A008648 floor((n+4)/5), n>0.
%p A008648 1/(1-x)/(1-x^5)/(1-x^25)
%Y A008648 Cf. A002266.
%Y A008648 Sequence in context: A105512 A002266 A075249 this_sequence A105511 A027868 A060384
%Y A008648 Adjacent sequences: A008645 A008646 A008647 this_sequence A008649 A008650 A008651
%K A008648 nonn,easy
%O A008648 0,6
%A A008648 njas
%I A105511
%S A105511 0,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,7,7,7,7,8,9,9,9,9,10,10,10,
%T A105511 10,10,11,11,11,11,12,13,13,13,13,14,15,15,15,15,16,16,16,16,16,17,17,
%U A105511 17,17,17,18,18,18,18,19,20,20,20,20,21,22,22,22,22,23,23,23,23,23,24
%N A105511 Number of times 1 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.
%C A105511 a(n) = #{k: A008963(k) = 1 and 0<=k<=n};
%C A105511 a(A105501(n)) = a(A105501(n) - 1) + 1;
%C A105511 n = a(n) + A105512(n) + A105513(n) + A105514(n) + A105515(n) + A105516(n) + A105517(n) + A105518(n) + A105519(n).
%Y A105511 Cf. A000030, A000045, A105501.
%Y A105511 Sequence in context: A002266 A075249 A008648 this_sequence A027868 A060384 A105564
%Y A105511 Adjacent sequences: A105508 A105509 A105510 this_sequence A105512 A105513 A105514
%K A105511 nonn,base
%O A105511 0,3
%A A105511 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 11 2005
%I A027868
%S A027868 0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,6,6,6,6,6,7,7,7,
%T A027868 7,7,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,12,12,12,12,12,13,13,13,13,
%U A027868 13,14,14,14,14,14,15,15,15,15,15,16,16,16,16,16,18,18,18,18,18,19
%N A027868 Number of trailing zeros in n!; highest power of 5 dividing n!.
%H A027868 E. W. Weisstein,
Link to a section of The World of Mathematics.
%F A027868 Floor[n/5] + floor[n/25] + floor[n/125] + floor[n/625] + ....
%F A027868 Sum [ n/5^i ] from i=1 to infinity.
%F A027868 a(n)=(n-A053824(n))/4
%t A027868 Table[t = 0; p = 5; While[s = Floor[n/p]; t = t + s; s > 0, p *= 5]; t, {n, 0, 100} ]
%Y A027868 Cf. A011371 and A054861 for analogues involving powers of 2 and 3.
%Y A027868 Cf. A112765.
%Y A027868 Sequence in context: A075249 A008648 A105511 this_sequence A060384 A105564 A025811
%Y A027868 Adjacent sequences: A027865 A027866 A027867 this_sequence A027869 A027870 A027871
%K A027868 nonn,base,nice,easy
%O A027868 0,11
%A A027868 Warut Roonguthai (warut822(AT)yahoo.com)
%I A060384
%S A060384 1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,
%T A060384 8,8,8,8,9,9,9,9,9,10,10,10,10,10,11,11,11,11,11,12,12,12,12,12,13,13,
%U A060384 13,13,14,14,14,14,14,15,15,15,15,15,16,16,16,16,16,17,17,17,17,17,18
%N A060384 Number of decimal digits of n-th Fibonacci number.
%F A060384 a(n) = ceiling(n*ln(tau)/ln(10)) +0 or +1 where tau is the golden ratio. - Benoit Cloitre (abmt(AT)wanadoo.fr), Oct 29 2002
%Y A060384 Cf. A000045, A022307, A001605, A060319-A060321, A050815.
%Y A060384 Sequence in context: A008648 A105511 A027868 this_sequence A105564 A025811 A034258
%Y A060384 Adjacent sequences: A060381 A060382 A060383 this_sequence A060385 A060386 A060387
%K A060384 base,nonn
%O A060384 0,8
%A A060384 Labos E. (labos(AT)ana.sote.hu), Apr 03 2001
%I A105564
%S A105564 0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,
%T A105564 8,8,8,8,8,9,9,9,9,10,10,10,10,10,11,11,11,11,11,12,12,12,12,13,13,13,
%U A105564 13,13,14,14,14,14,14,15,15,15,15,16,16,16,16,16,17,17,17,17,17,18,18
%N A105564 Number of blocks of exactly 4 Fibonacci numbers having equal length <= n.
%C A105564 a(n)/n --> 5 - 1/Log10((1+Sqrt(5))/2) = 0.215... .
%C A105564 a(n) = Sum(A105563(k): 1<=k<=n); a(n) = n - A105566(n);
%D A105564 Juergen Spilker, Die Ziffern der Fibonacci-Zahlen, Elemente der Mathematik 58 (Birkhaeuser 2003).
%Y A105564 Sequence in context: A105511 A027868 A060384 this_sequence A025811 A034258 A090663
%Y A105564 Adjacent sequences: A105561 A105562 A105563 this_sequence A105565 A105566 A105567
%K A105564 nonn
%O A105564 1,8
%A A105564 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 14 2005
%I A025811
%S A025811 1,0,1,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,5,4,5,5,6,
%T A025811 6,6,6,7,7,8,8,8,9,9,10,10,10,11,11,12,12,13,13,14,14,15,
%U A025811 15,16,16,17,17,18,19,19,20,20,21,22,22,23,23,24,25,26
%N A025811 Expansion of 1/((1-x^2)(1-x^5)(1-x^11)).
%Y A025811 Sequence in context: A027868 A060384 A105564 this_sequence A034258 A090663 A111890
%Y A025811 Adjacent sequences: A025808 A025809 A025810 this_sequence A025812 A025813 A025814
%K A025811 nonn
%O A025811 0,11
%A A025811 njas
%I A034258
%S A034258 1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,8,8,8,8,8,9,9,10,
%T A034258 10,10,10,11,11,12,12,12,12,12,12,13,13,13,14,14,15,15,15,15,15,15,16,
%U A034258 17,17,17,17,18,18,18,19,19,19,20,20,20,20,21,21,21,21,21,22,22,22
%N A034258 Write n! as a product of n numbers, n = k(1)*k(2)*...*k(n) with k(1)<=k(2)<=..., in all possible ways; a(n) = max value of k(1).
%C A034258 36, 49, 52, and 55 are not in this sequence. - Don Reble (djr(AT)nk.ca), Nov 29 2001
%C A034258 a(n) >= a(n-1). - Larry Reeves (larryr(AT)acm.org), Jan 06 2005
%D A034258 R. K. Guy and J. L. Selfridge, Factoring factorial n, Amer. Math. Monthly, 105 (1998), 766-767.
%e A034258 3! = 6 = 1*2*3 is the only possible factorization, so a(3) = 1.
%e A034258 27! = 8^4 * 9^6 * 10^6 * 11^2 * 12 * 13^2 * 14^3 * 17 * 19 * 23, with 4 + 6 + 6 + 2 + 1 + 2 + 3 + 1 + 1 + 1 = 27 factors, which is the required number. Since the first factor is 8, a(27) >= 8. In fact no larger value can be obtained, and a(27) = 8.
%Y A034258 Cf. A034259, A034260.
%Y A034258 Sequence in context: A060384 A105564 A025811 this_sequence A090663 A111890 A104277
%Y A034258 Adjacent sequences: A034255 A034256 A034257 this_sequence A034259 A034260 A034261
%K A034258 nonn,nice
%O A034258 1,4
%A A034258 njas
%E A034258 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), May 12 2001
%I A090663
%S A090663 2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,8,8,8,8,9,9,9,
%T A090663 9,9,10,10,10,10,10,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,13,14,
%U A090663 14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,17,17,17,17,17,17,18,18
%N A090663 Second term in continued fraction for the n-th root of n.
%C A090663 The number of n's is: 5,4,4,5,4,5,5,5,5,5,5,6,5,6,5,6,6,5,6,6,6,6,...,
%t A090663 Table[ ContinuedFraction[n^(1/n), 2][[ -1]], {n, 2, 83}] (from Robert G. Wilson v Dec 22 2003)
%o A090663 (PARI) f(n) = for(x=2,n,a=contfrac(x^(1/x));print1(a[2]","))
%Y A090663 Sequence in context: A105564 A025811 A034258 this_sequence A111890 A104277 A005857
%Y A090663 Adjacent sequences: A090660 A090661 A090662 this_sequence A090664 A090665 A090666
%K A090663 nonn
%O A090663 2,1
%A A090663 Cino Hilliard (hillcino368(AT)hotmail.com), Dec 15 2003
%I A111890
%S A111890 1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,
%T A111890 7,7,7,7,7,8,8,8,8,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,
%U A111890 11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,13,13,14,14,14,14,14,14
%N A111890 Number of numbers m <= n such that 0 equals the second digit after decimal point of square root of n in decimal representation.
%C A111890 For n>1: if A111862(n)=4 then a(n)=a(n-1)+1 else a(n)=a(n-1).
%C A111890 a(n)/n --> 1/10.
%D A111890 G. Polya and G. Szego, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Two, Chap. 4, Sect. 4, Problem 178.
%e A111890 a(10) = 3, a(100) = 15, a(1000) = 104, a(10000) = 1006.
%Y A111890 Cf. A111891, A111892, A111893, A111894, A111895, A111896, A111897, A111898, A111899, A111850.
%Y A111890 Sequence in context: A025811 A034258 A090663 this_sequence A104277 A005857 A025809
%Y A111890 Adjacent sequences: A111887 A111888 A111889 this_sequence A111891 A111892 A111893
%K A111890 nonn,base
%O A111890 1,4
%A A111890 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 20 2005
%I A104277
%S A104277 1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,6,7,7,8,8,8,10,10,11,11,13,13,14,
%T A104277 14,14,16,16,18,18,20,20,22,23,23,25,25,28,30,30,33,35,35,38,39,43,43,
%U A104277 46,46,49,51,51,55,56,60,61
%N A104277 Number of partitions of n in which both even and odd squares occur with multiplicity 1. There is no restriction on the parts which are twice odd squares.
%F A104277 Gf: product_{k>0}((1+x^(2k)^2))/(1-x^(2k-1)^2)).
%e A104277 E.g. a(21)=7 because we can write 21 as 18+2+1=16+4+1=16+2+2+1=9+4+2+2+2+2=9+2+2+2+2+2+2=4+2+2+2+2+2+2+2+2+1=2+2+2+2+2+2+2+2+2+2+1.
%p A104277 series(product((1+x^((2*k)^2))/(1-x^((2*k-1)^2)),k=1..100),x=0,100);
%Y A104277 Sequence in context: A034258 A090663 A111890 this_sequence A005857 A025809 A114575
%Y A104277 Adjacent sequences: A104274 A104275 A104276 this_sequence A104278 A104279 A104280
%K A104277 easy,nonn
%O A104277 0,5
%A A104277 Noureddine Chair (n.chair(AT)rocketmail.com), Mar 01 2005
%I A005857 M0216
%S A005857 1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,8
%N A005857 The coding-theoretic function A(n,12,7).
%D A005857 A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, New table of constant weight codes, IEEE Trans. Info. Theory 36 (1990), 1334-1380.
%H A005857 E. M. Rains and N. J. A. Sloane,
A(n,d,w) tables
%H A005857
Index entries for sequences related to A(n,d,w)
%Y A005857 Sequence in context: A090663 A111890 A104277 this_sequence A025809 A114575 A090735
%Y A005857 Adjacent sequences: A005854 A005855 A005856 this_sequence A005858 A005859 A005860
%K A005857 nonn,hard
%O A005857 7,7
%A A005857 njas
%E A005857 The version in the Encyclopedia of Integer Sequences had 1 instead of 2 at n=13.
%I A025809
%S A025809 1,0,1,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,5,4,5,5,6,6,6,
%T A025809 7,7,8,8,8,9,9,10,10,11,11,12,12,13,13,14,14,15,16,16,17,
%U A025809 17,18,19,19,20,20,22,22,23,23,24,25,26,26,27,28,29,30
%N A025809 Expansion of 1/((1-x^2)(1-x^5)(1-x^9)).
%Y A025809 Sequence in context: A111890 A104277 A005857 this_sequence A114575 A090735 A090736
%Y A025809 Adjacent sequences: A025806 A025807 A025808 this_sequence A025810 A025811 A025812
%K A025809 nonn
%O A025809 0,10
%A A025809 njas
%I A114575
%S A114575 1,1,2,2,2,2,2,3,3,3,4,3,2,4,2,4,4,1,5,2,3,2,4,2,3,3,3,4,3,3,2,5,5,3,7,
%T A114575 4,3,3,4,5,2,5,4,3,6,5,3,4,4,1,4,5,5,6,4,5,6,3,4,2,4,5,7,9,3,6,7,8,5,3,
%U A114575 5,7,5,5,7,3,5,6,6,6
%N A114575 Number of distinct prime factors of floor(e^n).
%e A114575 floor(e^3) = floor(20.08553) = 20. 20 has two distinct prime factors (2 and 5), therefore a(3) = 2.
%t A114575 Table[Length[FactorInteger[Floor[E^n]]], {n, 1, 80}]
%Y A114575 Cf. A001113 [decimal expansion of e].
%Y A114575 Sequence in context: A104277 A005857 A025809 this_sequence A090735 A090736 A094999
%Y A114575 Adjacent sequences: A114572 A114573 A114574 this_sequence A114576 A114577 A114578
%K A114575 nonn
%O A114575 0,3
%A A114575 Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Feb 17 2006
%I A090735
%S A090735 0,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,7,7,7,7,7,8,8,
%T A090735 8,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,12,12,12,
%U A090735 13,13,13,13,14,14,14,14,14,14,14,14,15,16,16,16,16,16,16,16,16,17,17
%N A090735 Number of positive square-free numbers <=n that can be expressed as a sum of 2 squares >0.
%D A090735 S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 100
%F A090735 a(n) is asymptotic to (6K/Pi^2)*n/sqrt(log(x)) where K is the Landau-Ramanujan constant
%o A090735 (PARI) a(n)=sum(i=1,n,issquarefree(i)*if(sum(u=1,i,sum(v=1,u,if(u^2+v^2-i,0,1))),1,0))
%Y A090735 Cf. A064533.
%Y A090735 Sequence in context: A005857 A025809 A114575 this_sequence A090736 A094999 A120202
%Y A090735 Adjacent sequences: A090732 A090733 A090734 this_sequence A090736 A090737 A090738
%K A090735 nonn
%O A090735 1,5
%A A090735 Benoit Cloitre (abmt(AT)wanadoo.fr), Jan 18 2004
%I A090736
%S A090736 0,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,9,9,
%T A090736 9,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,13,13,13,13,13,14,14,
%U A090736 14,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,18,18,19
%N A090736 Number of positive integers <=n that can be expressed as a sum of 2 coprime squares >0.
%D A090736 S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 100
%F A090736 a(n) is asymptotic to (3/8/K)*n/sqrt(log(n)) where K is the Landau-Ramanujan constant
%o A090736 (PARI) a(n)=sum(i=1,n,if(sum(u=1,i,sum(v=1,u,if(abs(u^2+v^2-i)+abs(gcd(u,v)-1),0,1))),1,0))
%Y A090736 Cf. A064533.
%Y A090736 Sequence in context: A025809 A114575 A090735 this_sequence A094999 A120202 A005861
%Y A090736 Adjacent sequences: A090733 A090734 A090735 this_sequence A090737 A090738 A090739
%K A090736 nonn
%O A090736 1,5
%A A090736 Benoit Cloitre (abmt(AT)wanadoo.fr), Jan 18 2004
%I A094999
%S A094999 1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,5,5,6,6,7,8,8,9,10,11,12,13,14,
%T A094999 16,17,19,21,22,25,27,29,32,35,38,42,45,50,54,59,65,71,77,84,92,100,109,
%U A094999 119,130,142,155,169,185,201,220,240,262,285,311,340,371,404,441,481
%N A094999 [ 12^n / 11^n ].
%t A094999 Table[ Floor[(12/11)^n], {n, 0, 71}]
%Y A094999 Cf. A002379, A094969 - A094500.
%Y A094999 Sequence in context: A114575 A090735 A090736 this_sequence A120202 A005861 A025788
%Y A094999 Adjacent sequences: A094996 A094997 A094998 this_sequence A095000 A095001 A095002
%K A094999 easy,nonn
%O A094999 0,9
%A A094999 Robert G. Wilson v (rgwv(AT)rgwv.com), May 26 2004
%I A120202
%S A120202 1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5,5,6,6,7,8,9,10,11,12,13,15,
%T A120202 16,18,20,22,25,28,31,34,38,42,47,52,58,64,71,79,88,98,109,121,134,149,
%U A120202 166,184,205,227,253,281,312,347,385,428,476,528,587,652,725,805,895
%N A120202 a(n)=ceiling( sum_{i=1..n-1} a(i)/8), a(1)=1.
%t A120202 f[s_] := Append[s, Ceiling[Plus @@ s/9]]; Nest[f, {1}, 70] (* RGWv *)
%Y A120202 Cf. A072493, A112088, A072493, A011782, A073941, A072493, A120160, A120170, A120178, A120186, A120194.
%Y A120202 Sequence in context: A090735 A090736 A094999 this_sequence A005861 A025788 A071806
%Y A120202 Adjacent sequences: A120199 A120200 A120201 this_sequence A120203 A120204 A120205
%K A120202 nonn
%O A120202 1,11
%A A120202 Graeme McRae (g_m(AT)mcraefamily.com), Jun 10 2006
%E A120202 Edited and extended by RGWv (rgwv(at)rgwv.com), Jul 07 2006
%I A005861 M0215
%S A005861 1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,5,6,6,7
%N A005861 The coding-theoretic function A(n,14,9).
%D A005861 A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, New table of constant weight codes, IEEE Trans. Info. Theory 36 (1990), 1334-1380.
%H A005861 E. M. Rains and N. J. A. Sloane,
A(n,d,w) tables
%H A005861
Index entries for sequences related to A(n,d,w)
%Y A005861 Sequence in context: A090736 A094999 A120202 this_sequence A025788 A071806 A025781
%Y A005861 Adjacent sequences: A005858 A005859 A005860 this_sequence A005862 A005863 A005864
%K A005861 nonn,hard
%O A005861 9,8
%A A005861 njas
%I A025788
%S A025788 1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,7,8,
%T A025788 8,9,9,9,10,10,11,11,12,13,13,14,14,15,15,16,17,17,18,18,
%U A025788 19,20,21,22,22,23,23,24,25,26,27,27,28,29,30,31,32,33
%N A025788 Expansion of 1/((1-x)(1-x^7)(1-x^12)).
%Y A025788 Sequence in context: A094999 A120202 A005861 this_sequence A071806 A025781 A018119
%Y A025788 Adjacent sequences: A025785 A025786 A025787 this_sequence A025789 A025790 A025791
%K A025788 nonn
%O A025788 0,8
%A A025788 njas
%I A071806
%S A071806 0,0,0,0,2,2,2,2,2,3,3,4,4,4,4,4,5,6,7,7,7,8,9,9,9,9,9,9,9,9,9,9,10,11,
%T A071806 12,12,12,15,18,18,19,19,20,20,20,20,20,20,20,21,21,22,22,23,23,23,23,
%U A071806 23,23,24,24,24,25,26,26,26,26,28,28,29,29,29,29,29,30,31,31,31,31,31
%N A071806 Number of pairs (x,y) such that prime(x) + prime(y) = y*tau(x) + x*tau(y), 1<=x<=y<=n.
%o A071806 (PARI) for(n=1,130,print1(sum(i=1,n,sum(j=1,i,if(prime(i)+prime(j)-j*numdiv(i)-i*numdiv(j),0,1))),","))
%Y A071806 Cf. A000005.
%Y A071806 Sequence in context: A120202 A005861 A025788 this_sequence A025781 A018119 A120502
%Y A071806 Adjacent sequences: A071803 A071804 A071805 this_sequence A071807 A071808 A071809
%K A071806 nonn
%O A071806 1,5
%A A071806 Benoit Cloitre (abmt(AT)wanadoo.fr), Jun 06 2002
%I A025781
%S A025781 1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,5,5,6,6,6,7,7,8,8,9,10,10,
%T A025781 11,11,12,13,13,14,14,15,16,17,18,18,19,20,21,22,22,23,
%U A025781 24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,40,41,42
%N A025781 Expansion of 1/((1-x)(1-x^5)(1-x^12)).
%Y A025781 Sequence in context: A005861 A025788 A071806 this_sequence A018119 A120502 A099480
%Y A025781 Adjacent sequences: A025778 A025779 A025780 this_sequence A025782 A025783 A025784
%K A025781 nonn
%O A025781 0,6
%A A025781 njas
%I A018119
%S A018119 1,2,2,2,2,2,3,3,4,4,4,5,6,7,7,8,10,11,13,14,16,19,22,25,
%T A018119 28,32,37,43,49,56,64,74,85,98,112,128,148,169,195,223,
%U A018119 256,295,338,389,446,512,589,676,777,892,1024,1177,1352
%N A018119 Powers of fifth root of 2 rounded up.
%Y A018119 Sequence in context: A025788 A071806 A025781 this_sequence A120502 A099480 A025783
%Y A018119 Adjacent sequences: A018116 A018117 A018118 this_sequence A018120 A018121 A018122
%K A018119 nonn
%O A018119 0,2
%A A018119 njas
%I A120502
%S A120502 1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,5,6,6,7,8,8,8,8,8,8,8,9,10,10,11,12,12,
%T A120502 12,13,14,14,15,16,16,16,16,16,16,16,16,17,18,18,19,20,20,20,21,22,22,
%U A120502 23,24,24,24
%N A120502 Meta-fibonacci sequence a(n) with parameters s=3.
%D A120502 B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees, and Extremal Compact Codes, Electronic Journal of Combinatorics, 13 (2006), #R26, 13 pages.
%H A120502 C. Deugau and F. Ruskey,
Complete k-ary Trees and Generalized Meta-Fibonacci Sequences
%F A120502 If 1 <= n <= 4, a(n)=1. If n = 5, then a(n)=2. If n>5 then a(n)=a(n-3-a(n-1)) + a(n-4-a(n-2))
%F A120502 g.f.: A(z) = z * (1 - z^3) / (1 - z) * sum(prod(z^3 * (1 - z^(2 * [i])) / (1 - z^[i]), i=1..n), n=0..infinity), where [i] = (2^i - 1).
%p A120502 a := proc(n)
%p A120502 option remember;
%p A120502 if n <= 4 then return 1 end if;
%p A120502 if n <= 5 then return 2 end if;
%p A120502 return add(a(n - i - 2 - a(n - i)), i = 1 .. 2)
%p A120502 end proc
%Y A120502 Cf. A120513, A120524.
%Y A120502 Sequence in context: A071806 A025781 A018119 this_sequence A099480 A025783 A025780
%Y A120502 Adjacent sequences: A120499 A120500 A120501 this_sequence A120503 A120504 A120505
%K A120502 nonn
%O A120502 1,5
%A A120502 Frank Ruskey (http://www.cs.uvic.ca/~ruskey/) and Chris Deugau (deugaucj(AT)uvic.ca), Jun 20 2006
%I A099480
%S A099480 1,2,2,2,2,2,3,4,4,4,4,4,5,6,6,6,6,6,7,8,8,8,8,8,9,10,10,10,10,10,11,12,
%T A099480 12,12,12,12,13,14,14,14,14,14,15,16,16,16,16,16,17,18,18,18,18,18,19,
%U A099480 20,20,20,20,20,21,22,22,22,22,22,23,24,24,24,24,24,25,26,26,26,26,26
%N A099480 Count from 1, repeating 2n five times.
%C A099480 Could be called the Jones sequence of the knot 9_43, since the g.f. is the reciprocal of (a parameterisation of) the Jones polynomial for 9_43.
%F A099480 G.f.: 1/((1-x+x^2)(1-x-x^3+x^4))=1/(1-2x+2x^2-2x^3+2x^4-2x^5+x^6); a(n)=2a(n-1)-2a(n-2)+2a(n-3)-2a(n-4)+2a(n-5)-a(n-6); a(n)=-cos(pi*2n/3+pi/3)/6+sqrt(3)sin(pi*2n/3+pi/3)/18-sqrt(3)cos(pi*n/3+pi/6)/6 +sin(pi*n/3+pi/6)/2+(n+3)/3.
%Y A099480 Cf. A099479.
%Y A099480 Sequence in context: A025781 A018119 A120502 this_sequence A025783 A025780 A109697
%Y A099480 Adjacent sequences: A099477 A099478 A099479 this_sequence A099481 A099482 A099483
%K A099480 easy,nonn
%O A099480 0,2
%A A099480 Paul Barry (pbarry(AT)wit.ie), Oct 18 2004
%I A025783
%S A025783 1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,5,6,6,6,6,7,8,9,9,9,
%T A025783 9,10,11,12,12,12,13,14,15,16,16,16,17,18,19,20,20,21,22,
%U A025783 23,24,25,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39
%N A025783 Expansion of 1/((1-x)(1-x^6)(1-x^11)).
%Y A025783 Sequence in context: A018119 A120502 A099480 this_sequence A025780 A109697 A103373
%Y A025783 Adjacent sequences: A025780 A025781 A025782 this_sequence A025784 A025785 A025786
%K A025783 nonn
%O A025783 0,7
%A A025783 njas
%I A025780
%S A025780 1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,5,6,6,6,6,7,8,9,9,9,10,11,
%T A025780 12,12,12,13,14,15,16,16,17,18,19,20,20,21,22,23,24,25,
%U A025780 26,27,28,29,30,31,32,33,34,35,37,38,39,40,41,43,44,45
%N A025780 Expansion of 1/((1-x)(1-x^5)(1-x^11)).
%Y A025780 Sequence in context: A120502 A099480 A025783 this_sequence A109697 A103373 A038539
%Y A025780 Adjacent sequences: A025777 A025778 A025779 this_sequence A025781 A025782 A025783
%K A025780 nonn
%O A025780 0,6
%A A025780 njas
%I A109697
%S A109697 1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,5,6,7,7,7,8,10,11,12,12,13,15,17,18,19,
%T A109697 20,23,26,28,29,31,34,38,41,43,45,50,55,60,63,66,71,79,85,90,94,101,110,
%U A109697 120,127,133,141,153,165,176,184,195,210,227,241,254,267,286,307,327
%N A109697 Number of partitions into parts each equal to 1 mod 5.
%F A109697 G.f.=1/product(1-x^(1+5j), j=0..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006
%e A109697 a(11)=3 since 11 = 11 = 6+1+1+1+1+1 = 1+1+1+1+1+1+1+1+1+1+1
%p A109697 g:=1/product(1-x^(1+5*j),j=0..25): gser:=series(g,x=0,85): seq(coeff(gser,x,n),n=0..80); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006
%Y A109697 Sequence in context: A099480 A025783 A025780 this_sequence A103373 A038539 A109368
%Y A109697 Adjacent sequences: A109694 A109695 A109696 this_sequence A109698 A109699 A109700
%K A109697 nonn
%O A109697 0,7
%A A109697 Erich Friedman (efriedma(AT)stetson.edu), Aug 07 2005
%E A109697 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006
%I A103373
%S A103373 1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,5,7,8,8,8,9,12,15,16,16,17,21,27,31,32,
%T A103373 33,38,48,58,63,65,71,86,106,121,128,136,157,192,227,249,264,293,349,
%U A103373 419,476,513,557,642,768,895,989,1070,1199,1410,1663,1884,2059,2269
%N A103373 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1 and for n>6: a(n) = a(n-5) + a(n-6).
%C A103373 k=5 case of the family of sequences whose k=1 case is the Fibonacci sequence A000045, k=2 case is the Padovan sequence A000931 (offset so as to begin 1,1,1), k=3 case is A079398 (offset so as to begin 1,1,1,1) and k=4 case is A103372.
%C A103373 The general case for integer k>1 is defined: a(1) = a(2) = ... = a(k+1) and for n>(k+1) a(n) = a(n-k) + a(n-[k+1]).
%C A103373 For this k=5 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^6 - x - 1 = 0. This is the real constant (to 100 digits accuracy): 1.134724138401519492605446054506472840279667226382801485925149551668236893999842671279689011614820249
%C A103373 The sequence of prime values in this k=5 case is A103383; The sequence of semiprime values in this k=5 case is A103393.
%D A103373 Selmer, E.S., "On the irreducibility of certain trinomials", Math. Scand., 4 (1956) 287-302
%D A103373 Shallit, J., "A generalization of automatic sequences", Theoretical Computer Science, 61(1988)1-16.
%D A103373 Zanten, A. J. van, "The golden ratio in the arts of painting, building, and mathematics", Nieuw Archief voor Wiskunde, 4 (17) (1999) 229-245.
%H A103373 Richard Padovan,
Dom Hans van der Laan and the Plastic Number.
%H A103373 J.-P. Allouche and T. Johnson,
Narayana's Cows and Delayed Morphisms
%e A103373 a(22) = 9 because a(22) = a(22-5) + a(22-6) = a(17) + a(16) = 5 + 4 = 9.
%t A103373 k = 5; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 65]
%Y A103373 Cf. A000045, A000931, A079398, A103372-A103381, A103383, A103393.
%Y A103373 Sequence in context: A025783 A025780 A109697 this_sequence A038539 A109368 A046774
%Y A103373 Adjacent sequences: A103370 A103371 A103372 this_sequence A103374 A103375 A103376
%K A103373 nonn,easy
%O A103373 1,7
%A A103373 Jonathan Vos Post (jvospost2(AT)yahoo.com), Feb 03 2005
%E A103373 Edited by Ray Chandler (RayChandler(AT)alumni.tcu.edu) and Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 06 2005
%I A038539
%S A038539 1,0,1,1,0,1,1,1,2,2,2,2,2,3,4,4,4,5,5,5,8,8,7,12,12,9,13,15,15,19,
%T A038539 21,21,23,25,28,34,35,37,45,45,45,56,59,61,77,80,76,92,100,101,119,
%U A038539 130,133,147,159,170,188,200,216,243,251,260,298,317,329,379,400
%N A038539 Complex semisimple Lie algebras of dimension n.
%C A038539 Direct consequence of classification of complex finite-dimensional simple Lie algebras
%D A038539 N. Jacobson, Lie Algebras, Dover Publications.
%F A038539 G.f.: (1+x)/((1 - x^14)(1 - x^52)(1 - x^78)(1-x^133)(1 - x^248) prod( 1-x^(n^2 + 2n), n = 1..inf) prod(1 - x^(2n^2 + n), n=2..inf) prod(1-x^(2n^2+n), n=3..inf) prod( 1-x^(2n^2 - n), n=4..inf)).
%Y A038539 Sequence in context: A025780 A109697 A103373 this_sequence A109368 A046774 A029105
%Y A038539 Adjacent sequences: A038536 A038537 A038538 this_sequence A038540 A038541 A038542
%K A038539 nonn,easy,nice
%O A038539 1,9
%A A038539 Paolo Dominici (pl.dm(AT)libero.it)
%I A109368
%S A109368 1,1,1,1,1,2,2,2,2,2,3,4,4,5,5,6,7,8,9,10,11,12,14,16,18,20,22,24,27,30,
%T A109368 34,37,40,44,49,54,60,65,71,78,85,94,103,112,122,132,144,158,172,186,
%U A109368 201,218,237,258,279,302,326,352,381,412,445,480,516,556,599,646
%N A109368 Expansion of q^(-1/2)eta(q^2)*eta(q^3)*eta(q^7)*eta(q^42)/(eta(q)*eta(q^6)*eta(q^14)*eta(q^21)) in powers of q.
%F A109368 Euler transform of period 42 sequence [1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, ...].
%F A109368 Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=v^2+v^3-v*w^2-v*u^2+v^2*w^2+v^2*u^2-u^2*w^2-v*u^2*w^2.
%F A109368 G.f.: Product_{k>0} (1+x^k)(1+x^(21k))/(1+x^(3k))(1+x^(7k))) = Product_{k>0} P42(x^k) where P42 is the 42nd cyclotomic polynomial.
%o A109368 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^3+A)*eta(x^7+A)*eta(x^42+A)/ (eta(x+A)*eta(x^6+A)*eta(x^14+A)*eta(x^21+A)), n))}
%Y A109368 Sequence in context: A109697 A103373 A038539 this_sequence A046774 A029105 A079954
%Y A109368 Adjacent sequences: A109365 A109366 A109367 this_sequence A109369 A109370 A109371
%K A109368 nonn
%O A109368 0,6
%A A109368 Michael Somos, Jun 26 2005
%I A046774
%S A046774 1,1,1,1,1,1,2,2,2,2,2,3,4,4,5,5,6,7,8,12,13,14,16,17,28,33,35,37,40,
%T A046774 61,77,83,87,94,132,168,186,194,213,277,350,392,414,460,569,703,793,
%U A046774 843,953,1139,1375,1550,1663,1894,2226,2628,2952,3187,3655,4249,4932
%N A046774 Number of partitions of n with equal number of parts congruent to each of 0, 2, 3 and 4 (mod 5).
%Y A046774 Sequence in context: A103373 A038539 A109368 this_sequence A029105 A079954 A079629
%Y A046774 Adjacent sequences: A046771 A046772 A046773 this_sequence A046775 A046776 A046777
%K A046774 nonn
%O A046774 0,7
%A A046774 David W. Wilson (davidwwilson(AT)comcast.net)
%I A029105
%S A029105 1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,6,7,8,8,8,9,10,12,13,14,
%T A029105 15,16,18,19,20,21,22,24,26,28,30,32,34,36,38,40,42,44,
%U A029105 46,49,52,55,58,61,64,67,70,73,76,79,83,87,91,95,99,104
%N A029105 Expansion of 1/((1-x)(1-x^5)(1-x^11)(1-x^12)).
%Y A029105 Sequence in context: A038539 A109368 A046774 this_sequence A079954 A079629 A029116
%Y A029105 Adjacent sequences: A029102 A029103 A029104 this_sequence A029106 A029107 A029108
%K A029105 nonn
%O A029105 0,6
%A A029105 njas
%I A079954
%S A079954 0,1,2,2,2,2,2,3,4,5,6,7,8,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,
%T A079954 10,10,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,
%U A079954 32,33,34,35,36,37,38,39,40,41,42,42,42,42,42,42,42,42,42,42,42,42,42,42
%N A079954 Partial sums of A030301.
%F A079954 a(n) = (n-1-(2/3)*(4^e_4-1)-(-1)^e_2*(n-1-2*(4^e_4-1)))/2 where e_4=floor(log[4](n)) and e_2=floor(log[2](n))=floor(log[4](n^2)) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 22 2003
%Y A079954 Sequence in context: A109368 A046774 A029105 this_sequence A079629 A029116 A064770
%Y A079954 Adjacent sequences: A079951 A079952 A079953 this_sequence A079955 A079956 A079957
%K A079954 nonn
%O A079954 1,3
%A A079954 njas, Feb 22 2003
%I A079629
%S A079629 2,2,2,2,2,3,5,3,7,6,6,10,13,7,8,9,9,7,12,18,14,24,19,10,21,21,20,20,19,
%T A079629 22,19,24,24,27,25,30,27,23,34,29,21,35,38,30,32,30,33,36,33,30
%N A079629 Number of twin prime pairs between p^2 and q^2 where (p,q) is the n-th twin prime pair.
%C A079629 Conjecturally a(n) is always positive. It seems that a(n) might tend to infinity.
%e A079629 a(3)=2 because the third twin prime pair is (11,13) and there are 2 twin prime pairs between 121 and 169, namely (137,139) and (149,151).
%Y A079629 Cf. A057767.
%Y A079629 Sequence in context: A046774 A029105 A079954 this_sequence A029116 A064770 A060467
%Y A079629 Adjacent sequences: A079626 A079627 A079628 this_sequence A079630 A079631 A079632
%K A079629 easy,nonn
%O A079629 1,1
%A A079629 Paul Boddington (psb(AT)maths.warwick.ac.uk), Jan 30 2003
%I A029116
%S A029116 1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,6,8,8,8,8,9,11,14,14,
%T A029116 14,14,15,17,20,20,20,21,23,26,30,30,30,31,33,36,40,40,
%U A029116 41,43,46,50,55,55,56,58,61,65,70,71,73,76,80,85,91,92
%N A029116 Expansion of 1/((1-x)(1-x^6)(1-x^11)(1-x^12)).
%Y A029116 Sequence in context: A029105 A079954 A079629 this_sequence A064770 A060467 A083533
%Y A029116 Adjacent sequences: A029113 A029114 A029115 this_sequence A029117 A029118 A029119
%K A029116 nonn
%O A029116 0,7
%A A029116 njas
%I A064770
%S A064770 0,1,1,1,2,2,2,2,2,3,10,11,11,11,12,12,12,12,12,13,10,11,11,11,12,12,
%T A064770 12,12,12,13,10,11,11,11,12,12,12,12,12,13,20,21,21,21,22,22,22,22,
%U A064770 22,23,20,21,21,21,22,22,22,22,22,23,20,21,21,21,22,22,22,22,22,23
%N A064770 Replace each digit of n by the floor of its square root.
%C A064770 The graph of this sequence is fractal-like.
%H A064770 O. Gerard,
Fractal behavior of this sequence (1)
%H A064770 O. Gerard,
Fractal behavior of this sequence (2)
%e A064770 26 -> [1.414...][2.449...] -> 12, so a(26) = 12.
%t A064770 Table[ FromDigits[ Floor[ Sqrt[ IntegerDigits[ n]]]], {n, 0, 100} ]
%Y A064770 Sequence in context: A079954 A079629 A029116 this_sequence A060467 A083533 A076500
%Y A064770 Adjacent sequences: A064767 A064768 A064769 this_sequence A064771 A064772 A064773
%K A064770 base,nonn,nice
%O A064770 0,5
%A A064770 Santi Spadaro (spados(AT)katamail.com), Oct 19 2001
%I A060467
%S A060467 0,1,1,1,2,2,2,2,2,3,11,2,1626,2,3,3,3,16,2,3,3,3,3,3
%V A060467 0,1,1,1,2,2,2,2,2,3,-11,2,1626,2,3,3,3,16,2,3,3,3,3,3
%N A060467 Consider solutions to n = x^3 + y^3 + z^3 (for n not 4 or 5 mod 9) with 0 <= |x| <= |y| <= |z|; take solution with smallest |z| and smallest |y|; sequence give value of z.
%C A060467 Indexed by A060464.
%D A060467 R. K. Guy, Unsolved Problems in Number Theory, Section D5.
%H A060467 H. Mishima,
About n=x^3+y^3+z^3
%e A060467 For n=16 the smallest solution is 16 = (-511)^3 + (-1609)^3 + 1626^3, which gives the term 1626.
%Y A060467 Cf. A060465-A060466.
%Y A060467 Sequence in context: A079629 A029116 A064770 this_sequence A083533 A076500 A060594
%Y A060467 Adjacent sequences: A060464 A060465 A060466 this_sequence A060468 A060469 A060470
%K A060467 sign,nice,hard
%O A060467 0,5
%A A060467 njas, Apr 10 2001
%I A083533
%S A083533 1,2,2,2,2,2,4,2,2,2,2,4,2,2,4,4,2,2,2,2,4,2,2,2,2,4,2,4,2,6,2,2,2,4,4,
%T A083533 4,4,2,2,2,2,2,2,4,4,6,2,2,2,4,2,2,4,4,2,6,4,2,2,2,2,4,4,2,2,4,6,2,4,2,
%U A083533 2,4,4,2,2,4,4,2,2,2,2,4,6,2,10,2,4,4,2,2,4,2,2,4,4,2,6,4,2,2,4,6,4,2,4
%N A083533 First difference sequence of A002202. Difference between consecutive possible values for phi[n].
%F A083533 a(n)=A002202[n+1]-A02202[n]
%e A083533 12 and 16 are the 7th and 8th possible totient values
%e A083533 12=phi[13],16=phi[17],
%e A083533 while {13,14,15} are impossible ones;
%e A083533 thus 16-12=4=a(7)=A002202[8]-A002202[7].
%t A083533 t=Table[EulerPhi[w], {w, 1, 25000}]; u=Union[%]; Delete[u-RotateRight[u], 1]
%Y A083533 Cf. A000010, A002202, A005277, A083531-A083536, A005277.
%Y A083533 Sequence in context: A029116 A064770 A060467 this_sequence A076500 A060594 A104361
%Y A083533 Adjacent sequences: A083530 A083531 A083532 this_sequence A083534 A083535 A083536
%K A083533 nonn
%O A083533 1,2
%A A083533 Labos E. (labos(AT)ana.sote.hu), May 20 2003
%I A076500
%S A076500 1,2,2,2,2,2,4,2,2,2,2,6,2,2,2,4,2,2,4,2,4,6,4,2,2,2,2,2,1,5,4,4,2,6,4,
%T A076500 2,6,2,10,8,2,2,2,1,1,2,2,4,4,2,4,2,4,2,6,8,4,12,4,2,2,10,6,8,1,13,2,6,
%U A076500 4,2,4,2,2,2,2,2,2,2,2,4,4,6,2,2,4,2,4,6,2,12,4,6,6,6,8,2,5,3,24,8,4,4
%N A076500 Distance between natural sculptures.
%C A076500 The 'sculpture' of a positive integer n is the infinite vector (c[1], c[2], ...), where c[k] is the number of prime factors p of n (counted with multiplicity) such that n^(1/(k+1)) < p <= n^(1/k). A number is in sequence A076450 if its sculpture is not equal to the sculpture of any smaller number. This sequence contains the first differences of A076450.
%H A076500 Jon Perry,
Sculptures
%e A076500 The first 8 terms of A076450 are 1,2,4,6,8,10,12,16, so a(1)=1, a(2)=...=a(6)=2, and a(7)=4.
%t A076500 sculpt[1]={}; sculpt[n_] := Module[{fn, v, i}, fn=FactorInteger[n]; v=Table[0, {Floor[Log[fn[[1, 1]], n]]}]; For[i=1, i<=Length[fn], i++, v[[Floor[Log[fn[[i, 1]], n]]]]+=fn[[i, 2]]]; v]; For[n=1; nlist=slist={}, n<500, n++, sn=sculpt[n]; If[ !MemberQ[slist, sn], AppendTo[slist, sn]; AppendTo[nlist, n]]]; Drop[nlist, 1]-Drop[nlist, -1]
%Y A076500 Cf. A076450.
%Y A076500 Sequence in context: A064770 A060467 A083533 this_sequence A060594 A104361 A086876
%Y A076500 Adjacent sequences: A076497 A076498 A076499 this_sequence A076501 A076502 A076503
%K A076500 nonn
%O A076500 1,2
%A A076500 Jon Perry (perry(AT)globalnet.co.uk), Nov 08 2002
%E A076500 Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Nov 18 2002
%I A060594
%S A060594 1,1,2,2,2,2,2,4,2,2,2,4,2,2,4,4,2,2,2,4,4,2,2,8,2,2,2,4,2,4,2,4,4,2,4,
%T A060594 4,2,2,4,8,2,4,2,4,4,2,2,8,2,2,4,4,2,2,4,8,4,2,2,8,2,2,4,4,4,4,2,4,4,4,
%U A060594 2,8,2,2,4,4,4,4,2,8,2,2,2,8,4,2,4,8,2,4,4,4,4,2,4,8,2,2,4,4,2,4,2
%N A060594 Number of non-congruent solutions of x^2 == 1 mod n (square roots of unity mod n).
%C A060594 Sum(k=1,n,a(k)) appears to be asymptotic to C*n*Log(n) with C=0.6... - Benoit Cloitre (abmt(AT)wanadoo.fr), Aug 19 2002
%C A060594 a(q) = number of real characters modulo q. - Benoit Cloitre (abmt(AT)wanadoo.fr), Feb 02 2003
%C A060594 Also number of real Dirichlet characters modulo n and sum(k=1,n,a(k)) is asymptotic to (6/pi^2)*n*ln(n). - Steven Finch (Steven.Finch(AT)inria.fr), Feb 16 2006
%D A060594 G. Tenenbaum, "Introduction a la theorie analytique et probabiliste des nombres", Cours specialise, 1995, Collection SMF, p. 260
%H A060594 Steven Finch and Pascal Sebah,
Squares and Cubes Modulo n (arXiv: math.NT/0604465).
%H A060594 K. Matthews,
Solving the congruence x^2=a(mod m)
%F A060594 If q is the number of distinct odd primes dividing n (sequence A005087) then: if 8 divides n a(n) = 2^(q+2) = 2^(A005087(n) + 2); if n == 4 (mod 8) a(n) = 2^(q+1) = 2^(A005087(n) + 1); otherwise a(n) = 2^q = 2^(A005087(n)) - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 29 2001
%F A060594 a(n)=2^omega(n)/2 if n==+/-2 (mod 8), a(n)=2^omega(n) if n==+/-1, +/-3, 4 (mod 8), a(n)=2*2^omega(n) if n==0 (mod 8), where omega(n)=A001221(n). - Benoit Cloitre (abmt(AT)wanadoo.fr), Feb 02 2003
%F A060594 For n>=2 A046073(n) * A060594(n) = A000010(n) = phi(n) (This gives a formula for A046073(n) using the one in A060594(n) ). - Sharon Sela (sharonsela(AT)hotmail.com), Mar 09 2002
%e A060594 The four numbers 1^2, 3^2, 5^2, and 7^2 are congruent to 1 mod 8, so a(8)=4.
%o A060594 (PARI) a(n)=sum(i=1,n,if((i^2-1)%n,0,1))
%Y A060594 Cf. A005087.
%Y A060594 Cf. A046073, A000010.
%Y A060594 Sequence in context: A060467 A083533 A076500 this_sequence A104361 A086876 A066691
%Y A060594 Adjacent sequences: A060591 A060592 A060593 this_sequence A060595 A060596 A060597
%K A060594 nonn,mult
%O A060594 1,3
%A A060594 Jud McCranie (j.mccranie(AT)adelphia.net), Apr 11 2001
%I A104361
%S A104361 1,2,2,2,2,2,4,2,2,2,4,4,4,4,2,4,8,8,2,8,8,8,2,4,16,16,4,4,32,16,4,8,4,
%T A104361 8,2,8,16,8,32,8,16,16,32,2,8,16,4,4,4
%N A104361 Number of divisors of A104350(n) - 1.
%C A104361 a(n) = A000005(A104357(n)).
%H A104361 R. Zumkeller,
Products of largest prime factors of numbers <= n
%Y A104361 Cf. A104362, A104369, A064145.
%Y A104361 Sequence in context: A083533 A076500 A060594 this_sequence A086876 A066691 A064133
%Y A104361 Adjacent sequences: A104358 A104359 A104360 this_sequence A104362 A104363 A104364
%K A104361 nonn
%O A104361 2,2
%A A104361 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 06 2005
%I A086876
%S A086876 1,2,2,2,2,2,4,2,2,4,4,4,2,2,4,4,4,4,4,4,4,2,2,4,4,4,4,4,4,4,4,4,4,4,4,
%T A086876 4,6,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,4,6,2,4,4,4,4,4,4,6,4,6,6,6,6,6,2,2,
%U A086876 4,4,4,4,4,4,4,4,4,4,4,4,4,6,2,4,4,4,4,4,4,6,4,6,6,6
%N A086876 Run lengths in A071542.
%C A086876 All a(n) are even for n>1.
%C A086876 Records occur at positions { 1,2,7,37,122,... } which correspond to run start positions { 2,4,16,126,512,... } in A071542.
%o A086876 (PARI) e1(n)=sum(k=0,floor(log2(n)),bittest(n,k))
%o A086876 f(n)=local(c):c=0:while(n,n=n-e1(n):c=c+1):c
%o A086876 p=1:r=1:for(n=1,150,c=0:while(f(r) == p,r=r+1:c=c+1):p=f(r):print1(c","))
%Y A086876 Sequence in context: A076500 A060594 A104361 this_sequence A066691 A064133 A105674
%Y A086876 Adjacent sequences: A086873 A086874 A086875 this_sequence A086877 A086878 A086879
%K A086876 nonn,easy
%O A086876 1,2
%A A086876 Ralf Stephan (ralf(AT)ark.in-berlin.de), Aug 21 2003
%I A066691
%S A066691 2,2,2,2,2,4,2,4,2,2,4,4,4,2,2,4,2,4,4,2,4,2,4,4,2,2,4,4,4,4,4,4,2,4,2,
%T A066691 6,4,4,2,2,4,4,4,2,4,4,4,4,4,2,4,4,4,4,4,2,6,2,4,4,2,4,4,6,4,4,2,4,4,4,
%U A066691 4,2,4,4,4,2,6,2,4,6,2,2,8,4,4,4,4,4,4,4,4,4,4,4,4,2,4,4,2,2,6,4,2,4,2
%N A066691 Value of tau(2n-1) when tau(2n-1)=tau(2n+1).
%o A066691 (PARI) for (n=2, 10000, if (numdiv(2*n-1) == numdiv(2*n+1), write1("tau=tau.txt", numdiv(2*n-1), ", ")))
%Y A066691 Sequence in context: A060594 A104361 A086876 this_sequence A064133 A105674 A001299
%Y A066691 Adjacent sequences: A066688 A066689 A066690 this_sequence A066692 A066693 A066694
%K A066691 nonn
%O A066691 0,1
%A A066691 Jon Perry (perry(AT)globalnet.co.uk), Jan 11 2002
%I A064133
%S A064133 2,2,2,2,2,4,4,4,4,2,4,2,2,8,4,2,8,4,8,4,4,2,8,4,4,2,8,4,16,8,16,2,8,8,
%T A064133 4,32,8,8,4,16,8,8,4,2,4,2,16,2,16,4,8,16,8,16,16,8,16,8,4,2,4,4,2,8,8,
%U A064133 4,32
%N A064133 Number of divisors of 6^n + 1 that are relatively prime to 6^m + 1 for all 0 < m < n.
%H A064133 Sam Wagstaff, Cunningham Project,
Factorizations of 6^n-1, n odd, n<330
%t A064133 a = {1}; Do[ d = Divisors[ 6^n + 1 ]; l = Length[ d ]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[ a, d[ [ k ] ] ] ] == {1}, c++ ]; k++ ]; Print[ c ]; a = Union[ Flatten[ Append[ a, Transpose[ FactorInteger[ 6^n + 1 ] ][ [ 1 ] ] ] ] ], {n, 0, 66} ]
%Y A064133 Sequence in context: A104361 A086876 A066691 this_sequence A105674 A001299 A001300
%Y A064133 Adjacent sequences: A064130 A064131 A064132 this_sequence A064134 A064135 A064136
%K A064133 nonn
%O A064133 0,1
%A A064133 Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 10 2001
%I A105674
%S A105674 2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,8,6,8,8,8,8,8,10,10,10,10,10
%N A105674 Highest minimal distance of any Type I (strictly) singly-even binary self-dual code of length 2n.
%D A105674 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977.
%H A105674 G. Nebe, E. M. Rains and N. J. A. Sloane,
Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
%H A105674 P. Gaborit,
Tables of Self-Dual Codes
%H A105674 E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998; (
Abstract,
pdf,
ps).
%e A105674 At length 8 the only strictly Type I self-dual code is {00,11}^4, which has d=2, so a(4) = 2.
%Y A105674 Cf. A105675, A105676, A105677, A105678, A016729, A066016, A105681, A105682.
%Y A105674 Cf. also A105685 for the number of such codes.
%Y A105674 Sequence in context: A086876 A066691 A064133 this_sequence A001299 A001300 A001306
%Y A105674 Adjacent sequences: A105671 A105672 A105673 this_sequence A105675 A105676 A105677
%K A105674 nonn,nice
%O A105674 1,1
%A A105674 njas, May 06 2005
%E A105674 The sequence continues: a(28) = either 10 or 12, then a(58) = 10, a(60) through a(68) = 12, ...
%I A001299
%S A001299 1,1,1,1,1,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,9,9,9,9,9,13,13,
%T A001299 13,13,13,18,18,18,18,18,24,24,24,24,24,31,31,31,31,31,
%U A001299 39,39,39,39,39,49,49,49,49,49,60,60,60,60,60,73,73,73
%N A001299 Number of ways of making change for n cents using coins of 1, 5, 10, 25 cents.
%D A001299 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 316.
%D A001299 G. P\'{o}lya and G. Szeg\"{o}, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1.
%H A001299 INRIA Algorithms Project,
Encyclopedia of Combinatorial Structures 175
%H A001299
Index entries for sequences related to making change.
%p A001299 1/(1-x)/(1-x^5)/(1-x^10)/(1-x^25)
%t A001299 CoefficientList[ Series[ 1 / ((1 - x)(1 - x^5)(1 - x^10)(1 - x^25)), {x, 0, 65} ], x ]
%Y A001299 Sequence in context: A066691 A064133 A105674 this_sequence A001300 A001306 A108105
%Y A001299 Adjacent sequences: A001296 A001297 A001298 this_sequence A001300 A001301 A001302
%K A001299 nonn
%O A001299 1,6
%A A001299 njas
%I A001300
%S A001300 1,1,1,1,1,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,9,9,9,9,9,13,13,
%T A001300 13,13,13,18,18,18,18,18,24,24,24,24,24,31,31,31,31,31,
%U A001300 39,39,39,39,39,50,50,50,50,50,62,62,62,62,62,77,77,77
%N A001300 Number of ways of making change for n cents using coins of 1, 5, 10, 25, 50 cents.
%D A001300 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 316.
%D A001300 G. P\'{o}lya and G. Szeg\"{o}, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1.
%H A001300 INRIA Algorithms Project,
Encyclopedia of Combinatorial Structures 176
%H A001300
Index entries for sequences related to making change.
%p A001300 1/(1-x)/(1-x^5)/(1-x^10)/(1-x^25)/(1-x^50)
%t A001300 CoefficientList[ Series[ 1 / ((1 - x)(1 - x^5)(1 - x^10)(1 - x^25)(1 - x^50)), {x, 0, 65} ], x ]
%Y A001300 Sequence in context: A064133 A105674 A001299 this_sequence A001306 A108105 A063468
%Y A001300 Adjacent sequences: A001297 A001298 A001299 this_sequence A001301 A001302 A001303
%K A001300 nonn
%O A001300 1,6
%A A001300 njas
%I A001306
%S A001306 1,1,1,1,1,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,10,10,10,10,10,
%T A001306 14,14,14,14,14,20,20,20,20,20,26,26,26,26,26,35,35,35,
%U A001306 35,35,44,44,44,44,44,57,57,57,57,57,70,70,70,70,70,88
%N A001306 Number of ways of making change for n cents using coins of 1, 5, 10, 20, 50, 100 cents.
%D A001306 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 316.
%D A001306 G. P\'{o}lya and G. Szeg\"{o}, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1.
%H A001306 INRIA Algorithms Project,
Encyclopedia of Combinatorial Structures 179
%H A001306
Index entries for sequences related to making change.
%p A001306 1/(1-x)/(1-x^5)/(1-x^10)/(1-x^20)/(1-x^50)/(1-x^100)
%t A001306 CoefficientList[ Series[ 1 / ((1 - x)(1 - x^5)(1 - x^10)(1 - x^20)(1 - x^50)(1 - x^100)), {x, 0, 60} ], x ]
%Y A001306 Sequence in context: A105674 A001299 A001300 this_sequence A108105 A063468 A010336
%Y A001306 Adjacent sequences: A001303 A001304 A001305 this_sequence A001307 A001308 A001309
%K A001306 nonn
%O A001306 0,6
%A A001306 njas
%I A108105
%S A108105 1,1,1,1,1,2,2,2,2,2,4,4,4,4,4,8,8,8,8,8,16,16,16,16,16,32,32,32,32,32,
%T A108105 64,64,64,64,64,128,128,128,128,128,256,256,256,256,256,512,512,512,512,
%U A108105 512,1024
%N A108105 A slow 6-symbol substitution by lengths of steps with characteristic polynomial: x*(x^5-2).
%F A108105 1->{2}, 2->{3}, 3->{4}, 4->{5}, 5->{1, 6}, 6->{2}
%t A108105 s[1] = {2}; s[2] = {3}; s[3] = {4}; s[4] = {5}; s[5] = {1, 6}; s[6] = {2}; t[a_] := Flatten[s /@ a]; p[0] = {1}; p[1] = t[p[0]]; p[n_] := t[p[n - 1]] a0 = Table[Length[p[i]], {i, 0, 50}]
%Y A108105 Sequence in context: A001299 A001300 A001306 this_sequence A063468 A010336 A054537
%Y A108105 Adjacent sequences: A108102 A108103 A108104 this_sequence A108106 A108107 A108108
%K A108105 nonn,uned
%O A108105 0,6
%A A108105 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jun 03 2005
%I A063468
%S A063468 0,0,0,0,2,2,2,2,2,4,4,4,6,6,8,8,10,10,10,12,12,12,12,12,16,18,18,18,
%T A063468 20,22,22,22,22,24,26,26,28,28,30,32,34,34,34,34,36,36,36,36,36,40,42,
%U A063468 44,46,46,48,48,48,50,50,52,54,54,54,54,62,62,62,64,64,66,66,66,68,70
%N A063468 Number of Pythagorean triples in the range [1..n] i.e. the number of integer solutions to x^2 + y^2 = z^2 with 1 <= x,y,z <= n.
%Y A063468 Cf. A062775.
%Y A063468 a(n) = 2*partial sums of A046080(n).
%Y A063468 Sequence in context: A001300 A001306 A108105 this_sequence A010336 A054537 A029104
%Y A063468 Adjacent sequences: A063465 A063466 A063467 this_sequence A063469 A063470 A063471
%K A063468 nonn
%O A063468 1,5
%A A063468 Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 27 2001
%E A063468 Corrected and extended by Vladeta Jovovic (vladeta(AT)Eunet.yu), Jul 28 2001
%I A010336
%S A010336 1,1,1,1,2,2,2,2,2,4,4,4,6,8,12,16,26
%N A010336 Maximal size of binary code of length n and asymmetric distance 5.
%D A010336 T. Etzion, New lower bounds for asymmetric and unidirectional codes, IEEE Trans. Inform. Theory, 37 (1991), 1696-1705.
%D A010336 J. H. Weber, Bounds and Constructions for Binary Block Codes Correcting Asymmetric or Unidirectional Errors, Ph. D. Thesis, Tech. Univ. Delft, 1989.
%D A010336 J. H. Weber, C. de Vroedt and D. E. Boekee, Bounds and constructions for binary codes of length less than 24 and asymmetric distance less than 6, IEEE Trans. Inform. Theory, 34 (1988), 1321-1332.
%Y A010336 Cf. A010101, A010238, A054536.
%Y A010336 Sequence in context: A001306 A108105 A063468 this_sequence A054537 A029104 A085543
%Y A010336 Adjacent sequences: A010333 A010334 A010335 this_sequence A010337 A010338 A010339
%K A010336 nonn,nice,hard
%O A010336 1,5
%A A010336 njas, Apr 10 2000
%I A054537
%S A054537 1,1,1,1,2,2,2,2,2,4,4,4,6,8,12,16,26
%N A054537 Same as A010336.
%Y A054537 Sequence in context: A108105 A063468 A010336 this_sequence A029104 A085543 A083499
%Y A054537 Adjacent sequences: A054534 A054535 A054536 this_sequence A054538 A054539 A054540
%K A054537 dead
%O A054537 1,5
%I A029104
%S A029104 1,1,1,1,1,2,2,2,2,2,4,4,5,5,5,7,7,8,8,8,11,11,13,13,14,
%T A029104 17,17,19,19,20,24,24,27,27,29,33,34,37,37,39,44,45,49,
%U A029104 49,52,57,59,63,64,67,73,75,80,81,85,91,94,99,101,105,113
%N A029104 Expansion of 1/((1-x)(1-x^5)(1-x^10)(1-x^12)).
%Y A029104 Sequence in context: A063468 A010336 A054537 this_sequence A085543 A083499 A029103
%Y A029104 Adjacent sequences: A029101 A029102 A029103 this_sequence A029105 A029106 A029107
%K A029104 nonn
%O A029104 0,6
%A A029104 njas
%I A085543
%S A085543 1,1,2,2,2,2,2,4,4,8,8,8,4,2,8,10,8,8,8,12,8,24,8,4,18,8,24,16,12,8,12,
%T A085543 8,12,18,8,12,2,8,12,40,24,16,8,20,12,8,16,24,8,12,8,12,32,12,32,24,8,8,
%U A085543 24,48,8,32,24,8,32,4,56,4,24,8,48,8,8,8,48,128,4,2,24,24,36,4,32,8,48
%N A085543 Number of divisors of the partition numbers (A000041).
%Y A085543 Sequence in context: A010336 A054537 A029104 this_sequence A083499 A029103 A008737
%Y A085543 Adjacent sequences: A085540 A085541 A085542 this_sequence A085544 A085545 A085546
%K A085543 nonn
%O A085543 0,3
%A A085543 Jason Earls (jcearls(AT)cableone.net), Jul 03 2003
%I A083499
%S A083499 2,2,2,2,2,4,5,2,3,3,2,3,4,5,3,2,3,2,3,2,3,2,3,2,2,3,2,4,2,2,3,2,3,2,2,
%T A083499 4,6,2,5,2,3,2,3,2,2,3,2,6,3,4,3,4,3,5,3,4,3,4,2,4,3,8,4,6,3,4,5,3,2,6,
%U A083499 5,2,4,2,3,4,8,3,2,3,7,4,2,3,2,3,2,4,2,4,2,5,3,2,5,2,4,2,23,3,11,2,5,2,3,4,5,2,5,2,3,2,3,2,3,6,2,5,2,5,2,3,4,7,2,3,2,6,7
%N A083499 a(n) = A083498(n)/n.
%C A083499 This is finite because A083498(130) does not exist. - Sam Alexander (amnalexander(AT)yahoo.com), Oct 20 2003
%Y A083499 Cf. A083498.
%Y A083499 Sequence in context: A054537 A029104 A085543 this_sequence A029103 A008737 A089452
%Y A083499 Adjacent sequences: A083496 A083497 A083498 this_sequence A083500 A083501 A083502
%K A083499 base,easy,nonn,fini,full
%O A083499 1,1
%A A083499 Amarnath Murthy and Meenakshi Srikanth (amarnath_murthy(AT)yahoo.com), May 03 2003
%E A083499 More terms from Sam Alexander (amnalexander(AT)yahoo.com), Oct 20 2003
%I A029103
%S A029103 1,1,1,1,1,2,2,2,2,2,4,5,5,5,5,7,8,8,8,8,11,13,14,14,14,
%T A029103 17,19,20,20,20,24,27,29,30,30,34,37,39,40,40,45,49,52,
%U A029103 54,55,60,64,67,69,70,76,81,85,88,90,97,102,106,109,111
%N A029103 Expansion of 1/((1-x)(1-x^5)(1-x^10)(1-x^11)).
%Y A029103 Sequence in context: A029104 A085543 A083499 this_sequence A008737 A089452 A115101
%Y A029103 Adjacent sequences: A029100 A029101 A029102 this_sequence A029104 A029105 A029106
%K A029103 nonn
%O A029103 0,6
%A A029103 njas
%I A008737
%S A008737 0,0,0,0,0,0,1,2,2,2,2,2,4,6,6,6,6,6,9,12,12,12,12,12,16,
%T A008737 20,20,20,20,20,25,30,30,30,30,30,36,42,42,42,42,42,49,
%U A008737 56,56,56,56,56,64,72,72,72,72,72,81,90,90,90,90,90,100
%N A008737 floor(n/6)*ceil(n/6).
%Y A008737 Sequence in context: A085543 A083499 A029103 this_sequence A089452 A115101 A023569
%Y A008737 Adjacent sequences: A008734 A008735 A008736 this_sequence A008738 A008739 A008740
%K A008737 nonn
%O A008737 0,8
%A A008737 njas
%I A089452
%S A089452 2,2,2,2,2,5,3,2,3,5,2,5,2,2,2,3,2,2,2,5,3,113,3,5,3,2,29,3,2,2,3,2,5,3,
%T A089452 3,5,2,2,5,5,2,2,2,17,11,2,7,11,19,3,3,13,2,2,2,5,2,2,11,3,2,2,5,2,11,2,
%U A089452 2,2,5,3,3,19,2,5,5,3,5,2,19,5,2,2,3,2,5,17,2,7,2,3,2,2,3,5,3,2,2,11,2
%N A089452 a(n) = smallest prime k such that k*(p(n)-1) + p(n) is prime, where p(n) = n-th prime.
%C A089452 Does every prime appear in this sequence? - Gabriel Cunningham (gcasey(AT)mit.edu), Mar 27 2004
%e A089452 a(2)=2 because 2*(p(2)-1) + p(2) = 7, which is prime. a(7)=5 because 2*(p(7)-1) + p(7) = 49 and 3*(p(7)-1) + p(7) = 65, both of which are composite, but 5*(p(7)-1) + p(7) = 97, which is prime.
%o A089452 (PARI) diff2sqp2(n) = { forprime(q=3,n, forprime(p=3,n, y=(p-q)/(q-1); if(y==floor(y), if(isprime(y), print1(y",");break) ) ) ) }
%Y A089452 Sequence in context: A083499 A029103 A008737 this_sequence A115101 A023569 A051887
%Y A089452 Adjacent sequences: A089449 A089450 A089451 this_sequence A089453 A089454 A089455
%K A089452 easy,nonn
%O A089452 3,1
%A A089452 Cino Hilliard (hillcino368(AT)hotmail.com), Dec 28 2003
%E A089452 More terms from Gabriel Cunningham (gcasey(AT)mit.edu), Mar 27 2004
%I A115101
%S A115101 1,1,1,1,2,2,2,2,2,5,4,7,7,6
%N A115101 Number of distinct prime factors of L(n + F(n)) where F(n) is the Fibonacci number and L(n) is the Lucas number, and n >= 1.
%e A115101 If n=1 then L(1 + F(1)) = 3 (prime) and so the first term is 1.
%e A115101 If n=2 then L(2 + F(2)) = 4 = 2^2 and so the second term is also 1.
%e A115101 If n=3 then L(3 + F(3)) = 11 (prime) and so the third term is also 1.
%Y A115101 Cf. A000045, A000032, A115051.
%Y A115101 Sequence in context: A029103 A008737 A089452 this_sequence A023569 A051887 A112968
%Y A115101 Adjacent sequences: A115098 A115099 A115100 this_sequence A115102 A115103 A115104
%K A115101 more,nonn
%O A115101 0,5
%A A115101 Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Mar 02 2006
%I A023569
%S A023569 2,2,2,2,2,5,7,2,5,13,7,17,19,5,11,5,7,29,2,17,7,19,5,43,47,7,5,13,
%T A023569 53,11,31,2,67,17,73,37,11,5,41,17,11,89,47,19,97,7,13,11,7,113,23,
%U A023569 59,17,31,127,13,19,67,137,139,7,29,19,11,31,157,41,167,43,173,7,89
%N A023569 Greatest prime divisor of prime(n)-3.
%Y A023569 Sequence in context: A008737 A089452 A115101 this_sequence A051887 A112968 A104588
%Y A023569 Adjacent sequences: A023566 A023567 A023568 this_sequence A023570 A023571 A023572
%K A023569 nonn
%O A023569 1,1
%A A023569 Clark Kimberling (ck6(AT)evansville.edu)
%I A051887
%S A051887 2,2,2,2,2,5,17,11,11,11,2,23,7,43,19,3,5,2,7,3,61,53,2,41,47,2
%N A051887 Minimal and special 2k-Germain primes, where 2k is A002110, primorial number.
%C A051887 Minimal p sequence so that primorial*p+1 is also prime.
%C A051887 While p is in A005384, the Q(n)p+1 primes are in A005385(primorial-safe primes)
%F A051887 Analogous to or subset of A051686, where the even numbers are: 2, 6, ..., A002110(n), ...
%e A051887 a(25) is 47 because Q(25)*47+1 is also prime and minimal with this property: Q(25)*47+1=47*2305567963945518424753102147331756070+1 =108361694305439365963395800924592535291 is a minimal prime. The first 6 terms (2,2,2,2,2,5) correspond to first entries in A005384, A007693, A051645, A051647, A051653, A051654 respectively.
%Y A051887 A002110, A005384, A005385, A051686, A007693, A051886, A051888.
%Y A051887 Sequence in context: A089452 A115101 A023569 this_sequence A112968 A104588 A010671
%Y A051887 Adjacent sequences: A051884 A051885 A051886 this_sequence A051888 A051889 A051890
%K A051887 nonn
%O A051887 1,1
%A A051887 Labos E. (labos(AT)ana.sote.hu), Dec 15 1999
%I A112968
%S A112968 0,0,1,0,0,2,2,2,2,2,6,2,4,2,7,1,5,0,7,3,9,1,11,2,7,1,12,1,11,7,8,5,8,1,
%T A112968 18,3,10,1,13,1,7,13,12,2,13,6,16,3,11,3,15,4,16,13,15,4,15,4,17,11,14,
%U A112968 4,13,7,12,15,17,5,15,16,13,3,12,3,20,3,27,19,20,3,11,3
%V A112968 0,0,1,0,0,-2,-2,-2,-2,-2,-6,-2,-4,-2,-7,-1,-5,0,-7,-3,-9,1,-11,2,-7,1,-12,1,-11,7,-8,
%W A112968 -5,-8,-1,-18,3,-10,1,-13,1,-7,13,-12,-2,-13,6,-16,3,-11,3,-15,-4,-16,13,-15,-4,-15,4,
%X A112968 -17,11,-14,4,-13,7,-12,15,-17,-5,-15,16,-13,3,-12,3,-20,3,-27,19,-20,-3,-11,3
%N A112968 Sum(mu(i)*Omega(j): i+j=n), with mu=A008683 and Omega=A001222.
%e A112968 a(5)=mu(1)*Omega(4)+mu(2)*Omega(3)+mu(3)*Omega(2)+mu(4)*Omega(1) = 1*2 - 1*1 - 1*1 + 0*1 = 0.
%Y A112968 Cf. A013939, A112967, A068341, A112962, A112963, A112964, A112966.
%Y A112968 Sequence in context: A115101 A023569 A051887 this_sequence A104588 A010671 A088050
%Y A112968 Adjacent sequences: A112965 A112966 A112967 this_sequence A112969 A112970 A112971
%K A112968 sign
%O A112968 1,6
%A A112968 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Oct 07 2005
%E A112968 Corrected by njas, Mar 01 2006
%I A104588
%S A104588 1,1,1,2,2,2,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,30,30,30,30,30,30,30,
%T A104588 30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,210,210,210,210,210,
%U A104588 210,210,210,210,210,210,210,210,210,210,210,210,210,210,210,210,210
%N A104588 Product of primes less than or equal to sqrt(n).
%F A104588 For n>0, #p(n) appears {(p(n+1))^2 - (p(n))^2} times [from n=(p(n))^2 to n=(p(n+1))^2 - 1, inclusive], i.e. A002110(n) appears A069482(n) times [from n=A001248(n) to n=A001248(n+1)-1, inclusive]
%Y A104588 Sequence in context: A023569 A051887 A112968 this_sequence A010671 A088050 A058005
%Y A104588 Adjacent sequences: A104585 A104586 A104587 this_sequence A104589 A104590 A104591
%K A104588 nonn
%O A104588 1,4
%A A104588 Lekraj Beedassy (boodhiman(AT)yahoo.com), Mar 17 2005
%I A010671
%S A010671 1,1,1,1,1,2,2,2,2,2,6,8
%N A010671 Maximal size of binary code of length n correcting 4 unidirectional errors.
%D A010671 J. H. Weber, Bounds and Constructions for Binary Block Codes Correcting Asymmetric or Unidirectional Errors, Ph. D. Thesis, Tech. Univ. Delft, 1989.
%D A010671 J. H. Weber, C. de Vroedt and D. E. Boekee, Bounds and constructions for binary codes of length less than 24 and asymmetric distance less than 6, IEEE Trans. Inform. Theory, 34 (1988), 1321-1332.
%Y A010671 Sequence in context: A051887 A112968 A104588 this_sequence A088050 A058005 A095386
%Y A010671 Adjacent sequences: A010668 A010669 A010670 this_sequence A010672 A010673 A010674
%K A010671 nonn
%O A010671 1,6
%A A010671 njas
%I A088050
%S A088050 2,2,2,2,2,11,11,11,11,11,11,11,11,2,101,101,101,101,101,101,101,101,
%T A088050 1093,8599,9601,9901,9901,10301,9901,9901,9901,10301,111893,798799,
%U A088050 1003001,996001,1003001,997001,1003001,997001,1003001,997001,11021993
%N A088050 Primes arising as successive differences in A088049. a(n) = A088049(n+1)-A088049(n).
%Y A088050 Cf. A088049, A088051.
%Y A088050 Sequence in context: A112968 A104588 A010671 this_sequence A058005 A095386 A060359
%Y A088050 Adjacent sequences: A088047 A088048 A088049 this_sequence A088051 A088052 A088053
%K A088050 base,nonn
%O A088050 1,1
%A A088050 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 20 2003
%E A088050 More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Jun 27 2005
%I A058005
%S A058005 2,2,2,2,2,12,2,2,2,4,2,4,2,4,30,2,2,12,2,20,6,4,2,12,2,4,2,56,2,4,2,2,
%T A058005 6,4,14,4,2,4,2,20,2,84,2,8,90,4,2,12,2,4,6,8,2,12,2,8,6,4,2,24,2,4,6,
%U A058005 2,10,132,2,4,6,20,2,36,2,4,30,8,154,12,2,20,2,4,2,56,10,4,6,88,2,20
%N A058005 a(n)=GCD[n,C(2n,n)].
%e A058005 a(n)=2 for values like 1,2,3,4,5,7,.. a(n)=2n for values like 6,15,...,190
%Y A058005 A001405.
%Y A058005 Sequence in context: A104588 A010671 A088050 this_sequence A095386 A060359 A029665
%Y A058005 Adjacent sequences: A058002 A058003 A058004 this_sequence A058006 A058007 A058008
%K A058005 nonn
%O A058005 0,1
%A A058005 Labos E. (labos(AT)ana.sote.hu), Nov 13 2000
%I A095386
%S A095386 2,2,2,2,2,13,2,13,2,13,2,5,13,5,2,13,13,11,5,2,13,5,3,11,5,577,13,11,5,
%T A095386 577,2,5,13,5,13,7,11,19,5,577,2,7,13,17,5,577,3,37,11,29,13,5,577,577,
%U A095386 7,7,11,19,5,23,577,577,2,7,5,19,17,13,5,577,3,577,7,17,11,29,19,101,5
%N A095386 Largest prime factor of peak values of 3x+1 trajectory started at n.
%F A095386 a[n]=A006530[A025586(n)]
%e A095386 n=27: peak=9232=2.2.2.2.577, so a[27]=577;
%e A095386 More extensive search suggests that all primes may occur as largest prime factor of peak.
%Y A095386 Cf. A006530, A025586.
%Y A095386 Sequence in context: A010671 A088050 A058005 this_sequence A060359 A029665 A056993
%Y A095386 Adjacent sequences: A095383 A095384 A095385 this_sequence A095387 A095388 A095389
%K A095386 nonn
%O A095386 2,1
%A A095386 Labos E. (labos(AT)ana.sote.hu), Jun 14 2004
%I A060359
%S A060359 2,2,2,2,2,14,18,18,32,32,54,54,54,40,62,62,2,2,2,2,42,42,30,30,
%T A060359 72,72,44,44,44,42,42,42,42,42,96,96,96,96,126,126,142,142,142,
%U A060359 142,2,2,142,142,142,142,122,122,122,122,122,122,262,262,98,98
%N A060359 { Smallest prime > LCM{1,2,...,n} } - { largest prime < LCM{1,2,...,n} }.
%D A060359 Cf. A003418, A060357, A060358.
%p A060359 [seq(nextprime(A003418(n))-prevprime(A003418(n)), n=3..100)];
%Y A060359 Sequence in context: A088050 A058005 A095386 this_sequence A029665 A056993 A057331
%Y A060359 Adjacent sequences: A060356 A060357 A060358 this_sequence A060360 A060361 A060362
%K A060359 nonn
%O A060359 3,1
%A A060359 njas, Apr 01 2001
%I A029665
%S A029665 2,2,2,2,2,16,2,2,36,2,2,64,140,196,2,204,336,2,100,540,714,2,1254,2,
%T A029665 144,506,1210,2640,2,650,1716,2,196,2366,8008,2,2,256,1240,4200,10556,
%U A029665 20384,30888,37180,2,1496,5440,14756,30940,51272,68068,2,324,6936
%N A029665 Even numbers to the left of the central elements of the (2,1)-Pascal triangle A029653.
%Y A029665 Sequence in context: A058005 A095386 A060359 this_sequence A056993 A057331 A067089
%Y A029665 Adjacent sequences: A029662 A029663 A029664 this_sequence A029666 A029667 A029668
%K A029665 nonn,tabf
%O A029665 0,1
%A A029665 Mohammad K. Azarian (ma3(AT)evansville.edu)
%E A029665 More terms from James A. Sellers (sellersj(AT)math.psu.edu)
%I A056993
%S A056993 2,2,2,2,2,30,102,120,278,46,824,150,1534,30406,67234,70906,48594,62722
%N A056993 a(n) = smallest k >= 2 such that k^(2^n)+1 is prime.
%C A056993 Smallest base value yielding generalized Fermat primes. - Hugo Pfoertner (hugo(AT)pfoertner.org), Jul 01 2003
%C A056993 The first 5 terms correspond with the known (ordinary) Fermat primes. A probable candidate for the next entry is 62722^131072+1, discovered by Michael Angel in 2003. It has 628808 decimal digits. - Hugo Pfoertner (hugo(AT)pfoertner.org), Jul 01 2003
%H A056993 Yves Gallot,
Generalized Fermat Prime Search
%e A056993 The primes are 2^(2^0)+1=3, 2^(2^1)+1=5, 2^(2^2)+1=17, 2^(2^3)+1=257, 2^(2^4)+1=65537, 30^(2^5)+1, 102^(2^6)+1, ....
%t A056993 Do[p = 2^k; n = 2; cp = n^p + 1; While[ !PrimeQ[cp], n = n + 1; cp = n^p + 1]; Print[n], {k, 0, 17}] (from Lei Zhou (lzhou5(AT)emory.edu), Feb 21 2005)
%Y A056993 Cf. A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002.
%Y A056993 Cf. A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002.
%Y A056993 Cf. A019434 (Fermat primes).
%Y A056993 Sequence in context: A095386 A060359 A029665 this_sequence A057331 A067089 A090872
%Y A056993 Adjacent sequences: A056990 A056991 A056992 this_sequence A056994 A056995 A056996
%K A056993 hard,nonn
%O A056993 0,1
%A A056993 Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 06 2000
%E A056993 1534 from Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 30 2000
%E A056993 62722 from Jeppe Stig Nielsen (sequence(AT)jeppesn.dk), Aug 07 2005
%I A057331
%S A057331 2,2,2,2,2,89,1122659,19099919,85864769,26089808579,554688278429,4090932431513069
%N A057331 a(n) = smallest prime p such that the first n iterates of p under x->2x+1 are all primes.
%C A057331 Initial terms of A000040, A005384, A007700, A023272, A023302, A023330.
%C A057331 For n>10 a(n) == -1 (mod 2*3*5*11*13). - Farideh Firoozbakht (f.firoozbakht(AT)math.ui.ac.ir), Apr 24 2004
%H A057331 C. K. Caldwell,
Latest results about Cunningham Chains
%H A057331
Index entries for sequences related to primes in arithmetic progressions
%e A057331 a(5) = 89 because the numbers 89, 179, 359, 719, 1439, 2879 are all primes and 89 is the first number to have this property.
%t A057331 f[n_] := 2n + 1; k = 1; Do[ While[ Union[ PrimeQ[ NestList[ f, Prime[k], n]]] != {True}, k++ ]; Print[ Prime[k]], {n, 1, 9} ]
%Y A057331 See also A005602.
%Y A057331 Sequence in context: A060359 A029665 A056993 this_sequence A067089 A090872 A063473
%Y A057331 Adjacent sequences: A057328 A057329 A057330 this_sequence A057332 A057333 A057334
%K A057331 nonn,nice
%O A057331 0,1
%A A057331 Patrick De Geest (pdg(AT)worldofnumbers.com), Aug 15 2000.
%E A057331 More terms from Farideh Firoozbakht (f.firoozbakht(AT)math.ui.ac.ir), Apr 24 2004
%E A057331 a(13) (from the Caldwell link) sent by Peter Deleu, Hulste, Belgium, Nov 22, 2004
%I A067089
%S A067089 2,2,2,2,2,185,18461,1842626,183987603,1837682236,1999303871,
%T A067089 2000827643,2000777468,2000722020,2000673854,2000631711,2000594530,
%U A067089 2000561482,2000531914,2000505305,2000481231,2000459347,2000439367
%N A067089 Floor[X/Y] where X = concatenation of (2n), (2n-1), ... down to n+1 and Y = concatenation of n, n-1, n-2, ... down to 1.
%e A067089 a(6)= floor [ 121110987/654321] = [185.094146451053840546153951959359]=185.
%t A067089 f[n_] := (k = 1; x = y = "0"; While[k < n + 1, x = StringJoin[x, ToString[2n - k + 1]]; y = StringJoin[ToString[2k - 1], y]; k++ ]; Return[ Floor[10* ToExpression[x] / ToExpression[y]]] ); Table[ f[n], {n, 1, 25} ]
%Y A067089 Cf. A067088.
%Y A067089 Sequence in context: A029665 A056993 A057331 this_sequence A090872 A063473 A096859
%Y A067089 Adjacent sequences: A067086 A067087 A067088 this_sequence A067090 A067091 A067092
%K A067089 easy,base,nonn
%O A067089 0,1
%A A067089 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jan 07 2002
%E A067089 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 09 2002
%I A090872
%S A090872 2,2,2,2,2,7072833120
%N A090872 a(n) is the smallest number m greater than 1 such that m^2^k+1 for k=0,1,...,n are primes.
%C A090872 The first five terms of this sequence correspond to Fermat primes.
%e A090872 a(5)=7072833120 because 7072833120^2^k+1 for k=0,1,2,3,4,5 are primes.
%Y A090872 Cf. A019434, A000215.
%Y A090872 Sequence in context: A056993 A057331 A067089 this_sequence A063473 A096859 A005086
%Y A090872 Adjacent sequences: A090869 A090870 A090871 this_sequence A090873 A090874 A090875
%K A090872 nonn
%O A090872 0,1
%A A090872 Farideh Firoozbakht (f.firoozbakht(AT)sci.ui.ac.ir), Jan 31 2004
%I A063473
%S A063473 1,1,2,2,2,2,3,1,2,3,2,2,2,1,2,4,3,1,2,0,1,3,3,3,3,2,3,2,1,1,2,1,0,2,1,
%T A063473 3,4,3,2,4,4,4,3,1,2,1,0,2,1,1,0,2,3,3,4,4,5,5,5,3,3,1,1,2,1,3,2,1,2,4,
%U A063473 3,1,0,1,0,1,1,1,2,0,1,0,1,1,1,2,3,4,3,3,4,4,3,3,3,5,6,6,7,8,7,5,4,3,2
%V A063473 1,-1,-2,-2,-2,-2,-3,-1,-2,-3,-2,-2,-2,-1,-2,-4,-3,-1,-2,0,-1,-3,-3,-3,-3,-2,-3,-2,-1,
%W A063473 -1,-2,-1,0,-2,-1,-3,-4,-3,-2,-4,-4,-4,-3,-1,-2,-1,0,2,1,1,0,-2,-3,-3,-4,-4,-5,-5,-5,
%X A063473 -3,-3,-1,-1,-2,-1,-3,-2,-1,-2,-4,-3,-1,0,1,0,-1,-1,-1,-2,0,1,0,-1,-1,-1,-2,-3,-4,-3
%N A063473 M(2*n-1), where M(n) is Mertens' function (A002321): Sum_{1<=k<=n} mu(k), where mu = Moebius function (A008683).
%o A063473 (PARI) M(n)=sum(k=1,n,moebius(k)); j=[]; for(n=1,200,j=concat(j,M(2*n-1))); j
%Y A063473 Cf. A002321, A008683.
%Y A063473 Sequence in context: A057331 A067089 A090872 this_sequence A096859 A005086 A020649
%Y A063473 Adjacent sequences: A063470 A063471 A063472 this_sequence A063474 A063475 A063476
%K A063473 easy,sign
%O A063473 1,3
%A A063473 Jason Earls (jcearls(AT)cableone.net), Jul 27 2001
%I A096859
%S A096859 1,1,2,2,2,2,3,1,2,3,3,1,3,2,2,3,3,4,2,2,4,2,2,3,4,2,4,4,2,3,4,4,4,5,4,
%T A096859 3,5,4,4,4,2,5,3,4,4,4,4,2,4,3,4,6,5,5,4,5,5,4,4,2,4,5,3,4,4,3,5,4,5,3,
%U A096859 4,2,4,4,3,3,5,3,5,3,4,4,4,3,4,5,5,3,4,3,3,3,5,3,5,2,6,4,3,7,5,3,3,3,5
%N A096859 Function A062401[x]=phi[sigma[x]]=f[x] is iterated. Starting with n, a(n) is the count of distinct terms arising in trajectory; a(n)=t(n)+c(n)=t+c, where t=number of transient terms, c=number of recurrent terms [in the terminal cycle].
%e A096859 n=255: list={255,144,360,288,[432,480],432,.},t=transient=4,c=cycle=2, a[255]=t+c=6;
%e A096859 n=244: list={244,180,144,360,288,[432,480],432,..},t=5,c=2,a[244]=7.
%t A096859 fs[x_] :=EulerPhi[DivisorSigma[1, x]] itef[x_, len_] :=NestList[fs, x, len] Table[Length[Union[itef[2^w, 20]]], {w, 1, 256}] (len=20 at n<=256 is suitable)
%Y A096859 Cf. A062401, A062402, A095955, A096860-A096864.
%Y A096859 Sequence in context: A067089 A090872 A063473 this_sequence A005086 A020649 A067131
%Y A096859 Adjacent sequences: A096856 A096857 A096858 this_sequence A096860 A096861 A096862
%K A096859 nonn
%O A096859 1,3
%A A096859 Labos E. (labos(AT)ana.sote.hu), Jul 21 2004
%I A005086
%S A005086 1,2,2,2,2,3,1,3,2,3,1,3,2,2,3,3,1,3,1,3,3,2,1,4,2,3,2,2,1,4,1,3,2,3,2,
%T A005086 3,1,2,3,4,1,4,1,2,3,2,1,4,1,3,2,3,1,3,3,3,2,2,1,4,1,2,3,3,3,3,1,3,2,3,
%U A005086 1,4,1,2,3,2,1,4,1,4,2,2,1,4,2,2,2,3,2,4,2,2,2,2,2,4,1,2,2,3,1,4,1,4,4
%N A005086 Number of Fibonacci numbers 1,2,3,5,... dividing n.
%p A005086 restart: with(combinat): for n from 1 to 200 do printf(`%d,`,sum(floor(n/fibonacci(k))-floor((n-1)/fibonacci(k)), k=2..15)) od:
%Y A005086 Cf. A038663.
%Y A005086 Sequence in context: A090872 A063473 A096859 this_sequence A020649 A067131 A094915
%Y A005086 Adjacent sequences: A005083 A005084 A005085 this_sequence A005087 A005088 A005089
%K A005086 nonn
%O A005086 1,2
%A A005086 njas
%E A005086 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 19 2001
%I A020649
%S A020649 2,2,2,2,3,2,2,2,2,2,2,3,2,2,3,2,2,2,2,2,5,2,2,2,2,2,2,2,3,2,2,3,2,2,2,2,
%T A020649 2,2,3,2,2,2,2,5,5,2,3,2,2,2,2,2,2,2,2,2,2,2,2,3,2,2,2,2,2,2,2,2,7,2,5,2,
%U A020649 2,2,2,2,3,2,2,3,2,2,2,2,2,2,3,2,2,2,2,5,2,2,5,3,2,2,2,2,3,2,2,2,2,2,2,2
%N A020649 Least nonresidue of n.
%H A020649 Eric Weisstein's World of Mathematics,
Quadratic Nonresidue
%Y A020649 Sequence in context: A063473 A096859 A005086 this_sequence A067131 A094915 A081147
%Y A020649 Adjacent sequences: A020646 A020647 A020648 this_sequence A020650 A020651 A020652
%K A020649 nonn
%O A020649 3,1
%A A020649 David W. Wilson (davidwwilson(AT)comcast.net)
%I A067131
%S A067131 1,2,2,2,2,3,2,2,2,2,2,4,2,2,3,2,2,3,2,2,2,2,2,4,2,2,2,3,2,3,2,2,2,2,2,
%T A067131 4,2,2,2,3,2,3,2,2,3,2,2,4,2,2,2,2,2,3,2,3,2,2,2,6,2,2,2,2,2,3,2,2,2,2,
%U A067131 2,4,2,2,3,2,2,3,2,3,2,2,2,4,2,2,2,2,2,3,3,2,2,2,2,4,2,2,2,2,2,3,2
%N A067131 Number of elements in the largest set of divisors of n which are in arithmetic progression.
%e A067131 a(12) = 4 as the divisors of 12 are {1,2,3,4,6,12} and the maximal subset in arithmetic progression is {1,2,3,4}. a(15) = 3; the maximal set is {1,3,5}.
%t A067131 lap[s_] := Module[{}, l=Length[s]; If[l<2, Return[l]]; val=2; For[i=1, i
val, val=k]]]; val]; lap/@Divisors/@Range[1, 200]
%Y A067131 Cf. A067132.
%Y A067131 Sequence in context: A096859 A005086 A020649 this_sequence A094915 A081147 A083399
%Y A067131 Adjacent sequences: A067128 A067129 A067130 this_sequence A067132 A067133 A067134
%K A067131 easy,nonn
%O A067131 1,2
%A A067131 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jan 09 2002
%E A067131 Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Jan 15 2002
%I A094915
%S A094915 2,2,2,2,3,2,2,2,2,3,2,2,2,2,2,3,2,2,2,2,3,2,2,2,3,2,2,2,3,3,2,2,2,
%T A094915 2,2,3,2,2,2,2,3,2,2,2,2,2,3,2,2,2,2,3,2,2,2,3,2,2,2,3,3,2,2,2,3,2,
%U A094915 2,2,2,2,3,2,2,2,2,3,2,2,2,2,2,3,2,2,2,2,3,2,2,2,3,2,2,2,3,3,2,2,2
%N A094915 Omit the 1's from A090822.
%H A094915 F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and A. R. Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence [pdf, ps].
%Y A094915 Sequence in context: A005086 A020649 A067131 this_sequence A081147 A083399 A105561
%Y A094915 Adjacent sequences: A094912 A094913 A094914 this_sequence A094916 A094917 A094918
%K A094915 nonn
%O A094915 1,1
%A A094915 njas, Jun 18 2004
%I A081147
%S A081147 2,2,2,2,3,2,2,2,3,2,2,2,3,2,2,2,3,2,2,2,2,3,2,2,2,3,2,2,2,3,2,2,2,3,2,
%T A081147 2,2,2,3,2,2,2,3,2,2,2,3,2,2,2,3,2,2,2,2,3,2,2,2,3,2,2,2,3,2,2,2,3,2,2,
%U A081147 2,2,3,2,2,2,3,2,2,2,3,2,2,2,3,2,2,2,3,2,2,2,2,3,2,2,2,3,2,2,2,3,2,2,2
%N A081147 Differences of Beatty sequence for square root of 5.
%C A081147 Let S(0) = 2; obtain S(k) from S(k-1) by applying 2 ->2223, 3 -> 22223; sequence is S(0), S(1), S(2), ...
%F A081147 a(n)=floor((n+1)*sqrt(5))-floor(n*sqrt(5))
%t A081147 Flatten[ Table[ Nest[ Flatten[ # /. {2 -> {2, 2, 2, 3}, 3 -> {2, 2, 2, 2, 3}}] &, {2}, n], {n, 0, 4}]] (from Robert G. Wilson v (rgwv(AT)rgwv.com), May 07 2005)
%o A081147 (PARI) a(n)=floor((n+1)*sqrt(5))-floor(n*sqrt(5))
%Y A081147 Cf. A022839.
%Y A081147 Sequence in context: A020649 A067131 A094915 this_sequence A083399 A105561 A087133
%Y A081147 Adjacent sequences: A081144 A081145 A081146 this_sequence A081148 A081149 A081150
%K A081147 nonn
%O A081147 0,1
%A A081147 Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 16 2003
%I A083399
%S A083399 1,2,2,2,2,3,2,2,2,3,2,3,2,3,3,2,2,3,2,3,3,3,2,3,2,3,2,3,2,4,2,2,3,3,3,
%T A083399 3,2,3,3,3,2,4,2,3,3,3,2,3,2,3,3,3,2,3,3,3,3,3,2,4,2,3,3,2,3,4,2,3,3,4,
%U A083399 2,3,2,3,3,3,3,4,2,3,2,3,2,4,3,3,3,3,2,4,3,3,3,3,3,3,2,3,3,3,2,4
%N A083399 Number of divisors of n that are not divisors of other divisors of n.
%C A083399 a(n)<=tau(n); a(n)=tau(n) iff n is prime or n=1 (A008578, A000040); a(n)=tau(n)-1 iff n is semiprime (A001358).
%F A083399 a(n)=omega(n)+1, where omega=A001221.
%e A083399 {1,2,3,4,6,8,12,24} are the divisors of n=24: 1, 2, 3, 4, and 6 divide not only 24, but also 8 or 12, therefore a(24)=3.
%Y A083399 Cf. tau=A000005.
%Y A083399 Complement of A055212.
%Y A083399 Sequence in context: A067131 A094915 A081147 this_sequence A105561 A087133 A062843
%Y A083399 Adjacent sequences: A083396 A083397 A083398 this_sequence A083400 A083401 A083402
%K A083399 nonn
%O A083399 1,2
%A A083399 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 12 2003
%I A105561
%S A105561 2,2,2,2,3,2,2,2,3,2,3,2,3,3,2,2,3,2,3,3,3,2,3,2,3,2,3,2,5,2,2,3,3,3,3,
%T A105561 2,3,3,3,2,5,2,3,3,3,2,3,2,3,3,3,2,3,3,3,3,3,2,5,2,3,3,2,3,5,2,3,3,5,2,
%U A105561 3,2,3,3,3,3,5,2,3,2,3,2,5,3,3,3,3,2,5,3,3,3,3,3,3,2,3,3,3
%N A105561 Primes whose indices are the number of distinct prime divisors of n.
%o A105561 (PARI) d(n) = for(x=2,n,print1(prime(omega(x))","))
%Y A105561 Sequence in context: A094915 A081147 A083399 this_sequence A087133 A062843 A033947
%Y A105561 Adjacent sequences: A105558 A105559 A105560 this_sequence A105562 A105563 A105564
%K A105561 easy,nonn,uned,obsc
%O A105561 2,1
%A A105561 Cino Hilliard (hillcino368(AT)hotmail.com), May 03 2005
%I A087133
%S A087133 1,2,2,2,2,3,2,2,2,3,2,3,2,3,3,2,2,3,2,4,3,3,2,3,2,3,2,4,2,4,2,2,3,3,3,
%T A087133 3,2,3,3,4,2,5,2,4,3,3,2,3,2,3,3,4,2,3,3,4,3,3,2,5,2,3,3,2,3,5,2,4,3,4,
%U A087133 2,3,2,3,3,4,3,5,2,4,2,3,2,6,3,3,3,5,2,4,3,4,3,3,3,3,2,3,4,4
%N A087133 Number of divisors of n that are not greater than the greatest prime-factor of n; a(1)=1.
%C A087133 a(n)<=A000005(n), a(n)=A000005(n) iff n is prime or n=1;
%C A087133 a(n)=2 iff n>1 is a prime power (A000961);
%C A087133 a(A087134(n))=n and a(k)
%H A087133 Eric Weisstein's World of Mathematics, Divisor Function
%H A087133 Eric Weisstein's World of Mathematics, Greatest Prime Factor
%e A087133 n=28: gpf(28)=7 and divisors = {1,2,4,7,14,28}: 1<=7, 2<=7, 4<=7, and 7<=7, therefore a(28)=4.
%Y A087133 Cf. A006530.
%Y A087133 Sequence in context: A081147 A083399 A105561 this_sequence A062843 A033947 A069719
%Y A087133 Adjacent sequences: A087130 A087131 A087132 this_sequence A087134 A087135 A087136
%K A087133 nonn
%O A087133 1,2
%A A087133 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 17 2003
%I A062843
%S A062843 1,1,2,2,2,2,3,2,2,2,3,2,3,3,4,2,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,2,2,3,
%T A062843 2,3,3,4,4,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,2,2,3,2,3,3,
%U A062843 4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3,4
%N A062843 Maximum number of ones in the representation of n in any base.
%e A062843 a[11]=3 since 11 in base 3 is 1011, containing 3 ones
%p A062843 seq(max(numboccur(convert(i,base,2),1), numboccur(convert(i,base,3),1), numboccur(convert(i,base,4),2), numboccur(convert(i,base,5),1), numboccur(convert(i,base,6),1), numboccur(convert(i,base,7),1), numboccur(convert(i,base,8),1), numboccur(convert(i,base,9),1), numboccur(convert(i,base,10),1), numboccur(convert(i,base,11),1), numboccur(convert(i,base,12),1), numboccur(convert(i,base,13),1), numboccur(convert(i,base,14),1), numboccur(convert(i,base,15),1), numboccur(convert(i,base,16),1), numboccur(convert(i,base,17),1)),i=1..200);
%Y A062843 Sequence in context: A083399 A105561 A087133 this_sequence A033947 A069719 A074592
%Y A062843 Adjacent sequences: A062840 A062841 A062842 this_sequence A062844 A062845 A062846
%K A062843 base,easy,nonn
%O A062843 1,3
%A A062843 Erich Friedman (efriedma(AT)stetson.edu), Jul 21 2001
%E A062843 Maple code and more terms from Barbara Haas Margolius (b.margolius(AT)csuohio.edu) Oct 10, 2001
%I A033947
%S A033947 1,1,2,2,2,2,3,2,2,2,3,2,6,3,2,2,2,2,2,2,5,2,2,3,5,2,2,2,6,3,3,2,3,2,2,
%T A033947 5,5,2,2,2,2,2,2,5,2,2,2,3,2,6,3,2,7,3,3,2,2,2,5,3,3,2,5,2,10,2,3,10,2,
%U A033947 2,3,2,2,2,2,2,2,5,3,21,2,2,5,5,5,2,3,13,2,2
%V A033947 1,-1,2,-2,2,2,3,2,-2,2,3,2,6,3,-2,2,2,2,2,-2,5,-2,2,3,5,2,-2,2,6,3,3,
%W A033947 2,3,2,2,-5,5,2,-2,2,2,2,-2,5,2,-2,2,3,2,6,3,-2,7,-3,3,-2,2,-2,5,3,3,
%X A033947 2,5,-2,10,2,3,10,2,2,3,-2,-2,2,2,-2,2,5,3,21,2,2,-5,5,-5,2,3,13,2,-2
%N A033947 Smallest primitive root (in absolute value) of n-th prime.
%Y A033947 Cf. A001918, A002199, A000040.
%Y A033947 Sequence in context: A105561 A087133 A062843 this_sequence A069719 A074592 A089993
%Y A033947 Adjacent sequences: A033944 A033945 A033946 this_sequence A033948 A033949 A033950
%K A033947 sign
%O A033947 1,3
%A A033947 Calculated by Jud McCranie (j.mccranie(AT)adelphia.net)
%I A069719
%S A069719 2,2,2,2,3,2,2,2,3,3,3,2,3,3,3,2,2,3,3,2,2,3,3,4,2,3,2,2,3,4,2,2,3,3,3,
%T A069719 2,2,3,4,3,2,3,3,2,3,2
%N A069719 Integer quotients arising in A068418: sum of proper divisors is divided by cototient of terms of A068418.
%F A069719 a(n)=A001065[A068418(n)]-A051953[A068418(n)]
%e A069719 n=39:m=A068418(39)=130141440, sigma[m]-m=405550080, m-Phi[m]=101387520,quotient=4=a(39)
%Y A069719 Cf. A068418, A001065, A000203, A051953, A068414, A069714, A069737.
%Y A069719 Sequence in context: A087133 A062843 A033947 this_sequence A074592 A089993 A047931
%Y A069719 Adjacent sequences: A069716 A069717 A069718 this_sequence A069720 A069721 A069722
%K A069719 nonn
%O A069719 1,1
%A A069719 Labos E. (labos(AT)ana.sote.hu), Apr 05 2002
%I A074592
%S A074592 2,2,2,2,3,2,2,3,2,2,2,2,2,3,2,5,2,2,3,2,2,2,3,2,2,2,3,2,2,5,2,3,2,2,2,
%T A074592 3,5,2,2,3,2,2,2,3,2,7,2,2,2,2,5,2,3,2,2,7,2,3,2,5,2,2,3,2,2,2,3,2,2,2,
%U A074592 3,2,2,5,2,3,2,7,2,2,3,2,2,3,2,2,7,2,3,2,2,2,3,2,11,2,5,2,3,2,2,2,3,2
%N A074592 Smallest prime factors of numbers that are not prime powers.
%C A074592 a(n)>2 iff n+1 and n+2 are prime powers (A006549).
%F A074592 a(n) = A020639(A024619(n)).
%Y A074592 Cf. A074593, A074594, A074595.
%Y A074592 Sequence in context: A062843 A033947 A069719 this_sequence A089993 A047931 A033618
%Y A074592 Adjacent sequences: A074589 A074590 A074591 this_sequence A074593 A074594 A074595
%K A074592 nonn
%O A074592 1,1
%A A074592 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 25 2002
%I A089993
%S A089993 2,2,2,2,3,2,2,3,2,2,2,2,3,3,2,5,2,2,3,2,3,2,3,2,2,2,3,2,2,5,2,3,2,3,2,
%T A089993 3,5,3,2,3,5,2,2,3,2,7,3,2,2,3,5,2,3,2,3,7,2,3,2,5,2,2,3,2,3,2,5,2,2,5,
%U A089993 3,2,3,5,2,3,2,7,3,2,3,2,3,3,5,3,7,2,3,2,3,5,3,2,11,2,5,2,3,2,3,2,3,7,5
%N A089993 Penultimate prime divisor of n if it exists.
%o A089993 (PARI) f(n) = a=factor(n);v=a[,1];ln=length(v);if(ln>1,return(v[ln-1])) g(m) = for(x=2,m,if(f(x)>0,print1(f(x)",")))
%Y A089993 Sequence in context: A033947 A069719 A074592 this_sequence A047931 A033618 A061357
%Y A089993 Adjacent sequences: A089990 A089991 A089992 this_sequence A089994 A089995 A089996
%K A089993 nonn
%O A089993 2,1
%A A089993 Cino Hilliard (hillcino368(AT)hotmail.com), Jan 14 2004
%I A047931
%S A047931 0,1,2,2,2,2,3,2,2,3,2,3,2,3,2,3,2,3,3,2,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,
%T A047931 3,3,2,3,3,2,3,3,3,2,3,3,3,2,3,3,3,2,3,3,3,2,3,3,3,3,2,3,3,3,2,3,3,3,3,
%U A047931 2,3,3,3,3,2,3,3,3,3,2,3,3,3,3,2,3,3,3,3,3
%N A047931 Number of new penny-penny contacts when putting pennies on a table following a spiral pattern.
%H A047931 R. W. Grosse-Kunstleve, Penny Spiral Sequence
%F A047931 The n-th "chunk" consists of 2 3{n-2} 2 3{n-1} 2 3{n-1} 2 3{n-1} 2 3{n-1} 2 3{n}, where a{b} symbolizes b repetitions of a.
%Y A047931 Cf. A047932.
%Y A047931 Sequence in context: A069719 A074592 A089993 this_sequence A033618 A061357 A120373
%Y A047931 Adjacent sequences: A047928 A047929 A047930 this_sequence A047932 A047933 A047934
%K A047931 nonn
%O A047931 1,3
%A A047931 Ralf W. Grosse-Kunstleve (rwgk(AT)cci.lbl.gov)
%I A033618
%S A033618 2,2,2,2,3,2,2,3,2,3,3,2,4,5,2,3,3,4,3,4,3,4,5,3,3,3,3,2,3,3,3,6,3,2,3,
%T A033618 2,3,3,3,3,4,2,3,3,7,3,4,3,4,5,4,3,4,2,2,3,3,3,4,3,3,3,3,4,3,2,3,6,2,3,3
%N A033618 Number of ways n-th repdigit number (A010785[ n ]) can be expressed as a polygonal number.
%H A033618 M. Keith, On Repdigit Polygonal Numbers, J. Integer Sequences, Vol. 1, 1998, #6.
%e A033618 The n-th k-sided polygonal number is P(n,k)=n((k-2)n+4-k)/2 (k >= 2, n >= 1). For each repdigit number R>=2, sequence gives number of (n,k) such that P(n,k)=R.
%Y A033618 Sequence in context: A074592 A089993 A047931 this_sequence A061357 A120373 A067595
%Y A033618 Adjacent sequences: A033615 A033616 A033617 this_sequence A033619 A033620 A033621
%K A033618 nonn
%O A033618 2,1
%A A033618 Mike Keith (domnei(AT)aol.com)
%I A061357
%S A061357 0,0,0,1,1,1,1,2,2,2,2,3,2,2,3,2,3,4,1,3,4,3,3,5,4,3,5,3,3,6,2,5,6,2,5,
%T A061357 6,4,5,7,4,4,8,4,4,9,4,4,7,3,6,8,5,5,8,6,7,10,6,5,12,3,5,10,3,7,9,5,5,
%U A061357 8,7,7,11,5,5,12,4,8,11,4,8,10,5,5,13,9,6,11,7,6,14,6,8,13,5,8,11,6,9
%N A061357 Number of 1<=k
%C A061357 Number of prime pairs (p,q) with p < n < q and q-n = n-p.
%C A061357 The same as the number of ways n can be expressed as the mean of two distinct primes.
%C A061357 Conjecture: for n>=4 a(n)>0. - Benoit Cloitre, Apr 29 2003
%C A061357 Conjectures from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jun 24 2003: 1) For each integer N>=1 there exists a positive integer m(N) such that for n>=m(N) a(n)>a(N). (After the first m(N)-1 terms, a(N) does not reappear). In particular, for N=1 (or 2 or 3), m(N)=4 and a(N)=0, giving Benoit Cloitre's conjecture. (cont.)
%C A061357 (cont.) Conjectures based upon observing a(1),...,a(10000):
%C A061357 m(4)=m(5)=m(6)=m(7)=m(19)=20 for a(4)=a(5)=a(6)=a(7)=a(19)=1,
%C A061357 m(8)=...(7 others)...=m(34)=35 for a(8)=...(7 others)...=a(34)=2,
%C A061357 m(12)=...(10 others)...=m(64)=65 for a(12)=...(10 others)...=a(64)=3,
%C A061357 m(18)=...(10 others)...=m(79)=80 for a(18)=...(10 others)...=a(79)=4,
%C A061357 m(24)=...(14 others)...=m(94)=95 for a(24)=...(14 others)...=a(94)=5,
%C A061357 m(30)=...(17 others)...=m(199)=200 for a(30)=...(17 others)...=a(199)=6, etc.
%C A061357 2) Each nonnegative integer appears at least once in the current sequence.
%C A061357 3) Stronger than 2): A001477 (nonnegative integers) is a subsequence of the current sequence. (Supporting evidence: I've observed that 0,1,2,...,175 is a subsequence of a(1),...,a(10000)).
%e A061357 a(10)= 2: there are two such pairs (3,17) and (7,13), as 10 = (3+17)/2 = (7+13)/2.
%Y A061357 Cf. A071681 (subsequence for prime n only).
%Y A061357 Sequence in context: A089993 A047931 A033618 this_sequence A120373 A067595 A074589
%Y A061357 Adjacent sequences: A061354 A061355 A061356 this_sequence A061358 A061359 A061360
%K A061357 nonn,easy
%O A061357 1,8
%A A061357 Amarnath_murthy (amarnath_murthy(AT)yahoo.com), Apr 28 2001
%E A061357 More terms from Larry Reeves (larryr(AT)acm.org), May 15 2001
%I A120373
%S A120373 0,0,0,1,1,1,1,2,2,2,2,3,2,2,3,2,3,4,1,3,4,3,3,5,4,3,5,3,3,6,2,5,6,2,5,
%T A120373 6,4,5,7,4,4,8,4,4,9,4,4,7,3,6,8,5,5,8,6,7,10,6,5,12,3,5,10,3,7,9,5,5,8,
%U A120373 7,7,11,5,5,12,4,8,11,4,8,10,5,5,13,9,6,11,7,6,14,6,8,13,5,8,11,6,9,13
%N A120373 Number of ways n^2 can be written as b^2+pq where 0
%C A120373 Apart from initial terms, same as A061357. The two sequences represent two different, although obviously related, scenarios. a(n) also gives the number of ways the even integer 2n can be written as the sum of two primes for all even integers >6. For example, a(8)=2 because 16=13+3=11+5. In this way, this sequence directly corresponds to the Goldbach Conjecture via the Panos-Noel Conjecture, which states that a(n)>0 for all n>3. If the Panos-Noel Conjecture is true, the Goldbach Conjecture immediately follows.
%D A120373 E. Noel and G. Panos, personal work, Jun 2006.
%H A120373 Weisstein, Eric W. "Goldbach Conjecture" From MathWorld--A Wolfram Web Resource.
%e A120373 a(8)=2 because 8^2=3^2+11*5=5^2+13*3
%o A120373 In Matlab: function pntif(y) w=[0 0 0]; for a=4:y t=[]; for b=1:a-2 r=a^2-b^2; s=factor(r); if length(s)==2 t=[t 1]; end end w=[w length(t)]; end w
%Y A120373 Cf. A061357.
%Y A120373 Sequence in context: A047931 A033618 A061357 this_sequence A067595 A074589 A081309
%Y A120373 Adjacent sequences: A120370 A120371 A120372 this_sequence A120374 A120375 A120376
%K A120373 nonn
%O A120373 1,8
%A A120373 Erin Noel and George Panos (erin.m.noel(AT)rice.edu), Jun 27 2006
%I A067595
%S A067595 0,1,1,2,2,2,2,3,2,2,3,3,3,3,4,3,3,3,4,3,3,5,4,4,4,5,3,3,4,4,4,4,6,5,5,
%T A067595 5,6,4,4,6,5,5,5,6,4,4,4,5,4,4,7,6,6,6,8,5,5,7,6,6,6,8,6,6,6,7,5,5,8,6,
%U A067595 6,6,7,4,4,5,5,5,5
%N A067595 Number of partitions of n into distinct Lucas parts (A000032).
%Y A067595 Sequence in context: A033618 A061357 A120373 this_sequence A074589 A081309 A010553
%Y A067595 Adjacent sequences: A067592 A067593 A067594 this_sequence A067596 A067597 A067598
%K A067595 easy,nonn
%O A067595 0,4
%A A067595 Naohiro Nomoto (n_nomoto(AT)yabumi.com), Jan 31 2002
%I A074589
%S A074589 2,2,2,2,3,2,2,5,5,2,2,7,13,7,2,2,11,29,29,11,2
%N A074589 Replace each number n in Pascal's triangle by the n-th prime.
%e A074589 Replacing each n in Pascal's triangle 1 1 1 1 2 1 1 3 3 1 .... by prime(n) gives a(n): 2 2 2 2 3 2 2 5 5 2 ....
%Y A074589 Sequence in context: A061357 A120373 A067595 this_sequence A081309 A010553 A108502
%Y A074589 Adjacent sequences: A074586 A074587 A074588 this_sequence A074590 A074591 A074592
%K A074589 easy,nonn
%O A074589 1,1
%A A074589 Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Sep 26 2002
%I A081309
%S A081309 0,0,2,2,2,2,3,2,3,2,2,3,5,2,3,7,5,2,3,2,3,13,5,23,7,2,3,19,2,3,7,5,17,
%T A081309 2,3,0,5,2,3,13,5,41,7,17,13,19,11,47,13,2,3,43,5,53,7,2,3,31,5,59,7,53,
%U A081309 31,37,11,2,3,41,5,43,7,71,19,2,3,67,5,0,7,53,17,73,2,3,13,5,23,7,17,89
%N A081309 Smallest prime p such that n-p is an 3-smooth number, a(n)=0 if no such prime exists.
%C A081309 a(n)=0 iff A081308(n)=0.
%e A081309 a(25)=7: 25=7+2*3^2.
%Y A081309 Cf. A081310, A081311, A081309, A000040, A003586.
%Y A081309 Sequence in context: A120373 A067595 A074589 this_sequence A010553 A108502 A078120
%Y A081309 Adjacent sequences: A081306 A081307 A081308 this_sequence A081310 A081311 A081312
%K A081309 nonn
%O A081309 1,3
%A A081309 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 17 2003
%I A010553
%S A010553 1,2,2,2,2,3,2,3,2,3,2,4,2,3,3,2,2,4,2,4,3,3,2,4,2,3,3,
%T A010553 4,2,4,2,4,3,3,3,3,2,3,3,4,2,4,2,4,4,3,2,4,2,4,3,4,2,4,
%U A010553 3,4,3,3,2,6,2,3,4,2,3,4,2,4,3,4,2,6,2,3,4,4,3,4,2,4,2
%N A010553 tau(tau(n)).
%D A010553 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%p A010553 with(numtheory): f := n->tau(tau(n));
%Y A010553 Cf. A000005.
%Y A010553 Sequence in context: A067595 A074589 A081309 this_sequence A108502 A078120 A057525
%Y A010553 Adjacent sequences: A010550 A010551 A010552 this_sequence A010554 A010555 A010556
%K A010553 nonn
%O A010553 1,2
%A A010553 njas
%I A108502
%S A108502 1,2,2,2,2,3,2,3,2,3,2,5,2,3,3,4,2,4,2,5,3,3,2,7,2,3,3,5,2,6,2,5,3,3,3,
%T A108502 7,2,3,3,7,2,6,2,5,4,3,2,10,2,4,3,5,2,6,3,7,3,3,2,11,2,3,4,6,3,6,2,5,3,
%U A108502 6,2,11,2,3,4,5,3,6,2,10,3,3,2,11,3,3,3,7,2,9,3,5,3,3,3,14,2,4,4,7,2,6
%N A108502 Number of factorizations of 4*n into distinct even numbers.
%e A108502 a(15)=3 because 15*4=60 can be factored as 60=30*2=10*6.
%Y A108502 Cf. A045778, A108501, A108503.
%Y A108502 Sequence in context: A074589 A081309 A010553 this_sequence A078120 A057525 A068211
%Y A108502 Adjacent sequences: A108499 A108500 A108501 this_sequence A108503 A108504 A108505
%K A108502 nonn
%O A108502 1,2
%A A108502 Christian G. Bower (bowerc(AT)usa.net), Jun 06 2005
%I A078120
%S A078120 0,1,1,2,2,2,2,3,2,3,2,3,3,3,3,3,3,4,2,3,3,3,3,4,4,3,3,4,4,3,3,4,4,3,3,
%T A078120 4,3,3,4,4,3,3,4,4,5,4,4,4,3,3,4,4,3,2,4,4,4,4,3,4,3,4,4,5,4,4,4,3,4,3,
%U A078120 4,4,5,4,4,4,4,4,4,5,3,4,5,4,5,5,4,5,4,4,4,5,4,4,5,4,4,3,5,5,4,4,5,4,4
%N A078120 Number of distinct prime divisors of n-th balanced number.
%F A078120 a(n)=A001221[A020492(n)]
%e A078120 575th balanced number is 2089542=2.3.7.13.43.89, a[575]=6, and sigma[2089542]/phi[2089542]=10.
%Y A078120 Cf. A020492, A001221.
%Y A078120 Sequence in context: A081309 A010553 A108502 this_sequence A057525 A068211 A089050
%Y A078120 Adjacent sequences: A078117 A078118 A078119 this_sequence A078121 A078122 A078123
%K A078120 nonn
%O A078120 1,4
%A A078120 Labos E. (labos(AT)ana.sote.hu), Dec 06 2002
%I A057525
%S A057525 1,1,2,2,2,2,3,2,3,2,3,3,3,3,4,3,3,3,4,3,3,3,4,3,4,3,4,4,4,4,5,3,4,3,4,
%T A057525 4,4,4,5,3,4,3,4,4,4,4,5,4,4,4,5,4,4,4,5,4,5,4,5,5,5,5,6,4,4,4,5,4,4,4,
%U A057525 5,4,5,4,5,5,5,5,6,4,4,4,5,4,4,4,5,4,5,4
%N A057525 Number of applications of f to reduce n to 1, where f(k) is the integer among k/2,(k+1)/4, (k+3)/4.
%e A057525 a(11)=3, which counts these reductions: 11->4->2->1.
%Y A057525 Sequence in context: A010553 A108502 A078120 this_sequence A068211 A089050 A054725
%Y A057525 Adjacent sequences: A057522 A057523 A057524 this_sequence A057526 A057527 A057528
%K A057525 nonn
%O A057525 2,3
%A A057525 Clark Kimberling (ck6(AT)evansville.edu), Sep 03 2000
%I A068211
%S A068211 2,2,2,2,3,2,3,2,5,2,3,3,2,2,2,3,3,2,3,5,11,2,5,3,3,3,7,2,5,2,5,2,3,3,
%T A068211 3,3,3,2,5,3,7,5,3,11,23,2,7,5,2,3,13,3,5,3,3,7,29,2,5,5,3,2,3,5,11,2,
%U A068211 11,3,7,3,3,3,5,3,5,3,13,2,3,5,41,3,2,7,7,5,11,3,3,11,5,23,3,2,3,7,5,5
%N A068211 Largest prime factor of Euler Phi of n.
%C A068211 Smallest numbers m, such that largest prime-factor of Phi[m] is prime(n), the n-th prime is also a prime number and identical to n-th term of A035095: Min[x; A068211(x)=prime(n)]=A035095(n); e.g. Phi[a(7)]=Phi[103]=2.3.17 of which 17=p(7) is the largest prime-factor.
%F A068211 a(n)=A006530[A000010(n)]
%e A068211 n=46, Phi[46]=2.2.11, a(46)=11
%t A068211 ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; Table[Max[ba[EulerPhi[w]]], {w, 1, 256}]
%Y A068211 Cf. A006530, A000010.
%Y A068211 Cf. A035095, A035096.
%Y A068211 Sequence in context: A108502 A078120 A057525 this_sequence A089050 A054725 A064415
%Y A068211 Adjacent sequences: A068208 A068209 A068210 this_sequence A068212 A068213 A068214
%K A068211 nonn
%O A068211 3,1
%A A068211 Labos E. (labos(AT)ana.sote.hu), Feb 21 2002
%I A089050
%S A089050 0,0,0,0,0,1,1,1,2,2,2,2,3,2,3,3,3,2,3,2,4,3,4,3,4,2,4,3,4,3,4,1,3,2,3,
%T A089050 2,5,3,4,3,5,3,5,3,5,3,4,1,4,2,4,3,5,3,4,1,4,3,4,1,4,1,1,0,3,2,3,2,5,3,
%U A089050 4,3,6,3,5,3,5,3,4,1,5,3,5,3,6,3,4,1,5,3,4,1,4,1,1,0,4,2,4,3,5,3,4,1,5
%N A089050 Number of ways of writing n as a sum of exactly 5 powers of 2.
%C A089050 The powers do not need to be distinct.
%Y A089050 A column of A089052. Cf. A036987, A075897, A089048, A089049, A089051, A089053.
%Y A089050 Sequence in context: A078120 A057525 A068211 this_sequence A054725 A064415 A086833
%Y A089050 Adjacent sequences: A089047 A089048 A089049 this_sequence A089051 A089052 A089053
%K A089050 nonn
%O A089050 0,9
%A A089050 njas, Dec 03 2003
%I A054725
%S A054725 1,1,1,2,2,2,2,3,2,3,3,3,3,3,3,4,4,3,3,4,3,4,4,4,4,4,3,4,4,4,4,5,4,5,4,
%T A054725 4,4,4,4,5,5,4,4,5,4,5,5,5,4,5,5,5,5,4,5,5,4,5,5,5,5,5,4,6,5,5,5,6,5,5,
%U A054725 5,5,5,5,5,5,5,5,5,6,4,6,6,5,6,5,5,6,6,5,5,6,5,6,5,6,6,5,5,6,6,6,6,6
%N A054725 a[1]=1; a[m]= sum[a[p-1]], where sum is over all primes (not necessarily distinct), p, that divide m.
%e A054725 a[20] = a[2-1] +a[2-1] +a[5-1] =1 + 1 +2 =4, because 20 = 2*2*5.
%Y A054725 Sequence in context: A057525 A068211 A089050 this_sequence A064415 A086833 A010764
%Y A054725 Adjacent sequences: A054722 A054723 A054724 this_sequence A054726 A054727 A054728
%K A054725 nonn
%O A054725 1,4
%A A054725 Leroy Quet (qq-quet(AT)mindspring.com), Apr 20 2000
%I A064415
%S A064415 0,1,1,2,2,2,2,3,2,3,3,3,3,3,3,4,4,3,3,4,3,4,4,4,4,4,3,4,4,4,4,5,4,5,4,
%T A064415 4,4,4,4,5,5,4,4,5,4,5,5,5,4,5,5,5,5,4,5,5,4,5,5,5,5,5,4,6,5,5,5,6,5,5,
%U A064415 5,5,5,5,5,5,5,5,5,6,4,6,6,5,6,5,5,6,6,5,5,6,5,6,5,6,6,5,5,6,6,6,6,6,5
%N A064415 a(n) = iter(n) if n is even, a(n) = iter(n)-1 if n is odd, where iter(n) = A003434(n) = smallest number of iterations of Euler totient function phi needed to reach 1.
%F A064415 For all integers m >0 and n>0 a(m*n)=a(m)+a(n). The function a(n) is completely additive. The smallest integer q which satisfy the equation a(q)=n is 2^q, the greatest is 3^q. For all integers n>0, the counter image off n, a^-1(n) is finite.
%Y A064415 Cf. A000010, A003434, A064416.
%Y A064415 Sequence in context: A068211 A089050 A054725 this_sequence A086833 A010764 A029383
%Y A064415 Adjacent sequences: A064412 A064413 A064414 this_sequence A064416 A064417 A064418
%K A064415 nonn
%O A064415 1,4
%A A064415 Christian WEINSBERG (cweinsbe(AT)fr.packardbell.org), Sep 30 2001
%E A064415 More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Jul 22 2002
%I A086833
%S A086833 1,1,1,2,2,2,2,3,2,3,3,3,3,3,3,4,4,3,3,4,3,4,5,4,4,4,3,4,4,4,4,5,5,5,4,
%T A086833 4,4,4,4,5,5,4,6,5,4,6,4,5,5,5,5,5,5,4,4,5,4,5,5,5,5,5,4,6,6,6,6,6,6,5,
%U A086833 5,5,5,5,5,5,7,5,5,6,4,6,7,5,6,7,5,6,6,5,5,7,5,5,5,6,6,6,6,6,6,6,6,6,5
%N A086833 Minimum number of different addends occurring in any shortest addition chain of Brauer type for a given n.
%C A086833 a(12509) is first undefined element of this sequence, because it is the smallest number that has no shortest addition chain of Brauer type. - Hugo Pfoertner (hugo(AT)pfoertner.org), Jun 10 2006
%H A086833 Giovanni Resta, Tables of Shortest Addition Chains, computed by David Wilson.
%e A086833 a(23)=5 because 23=1+1+2+1+4+9+5 is the shortest addition chain for 23.
%e A086833 For n=9 there are A079301(9)=3 different shortest addition chains, all of Brauer type:
%e A086833 [1 2 3 6 9] -> 9=1+1+1+3+3 -> 2 different addends {1,3}
%e A086833 [1 2 4 5 9] -> 9=1+1+2+1+4 -> 3 different addends {1,2,4}
%e A086833 [1 2 4 8 9] -> 9=1+1+2+4+1 -> 3 different addends {1,2,4}
%e A086833 The minimum number of addends is 2, therefore a(9)=2.
%Y A086833 Cf. A003064 A003065 A003313 A005766 A008057 A008928 A008933 A079300.
%Y A086833 Cf. A079300, A079301, A003313.
%Y A086833 Sequence in context: A089050 A054725 A064415 this_sequence A010764 A029383 A070098
%Y A086833 Adjacent sequences: A086830 A086831 A086832 this_sequence A086834 A086835 A086836
%K A086833 nonn
%O A086833 1,4
%A A086833 Tatsuru Murai (esatie(AT)mac.com), Aug 08 2003
%E A086833 Edited by Hugo Pfoertner (hugo(AT)pfoertner.org), Jun 10 2006
%I A010764
%S A010764 0,0,0,1,1,0,1,2,2,2,2,3,2,3,3,3,3,4,3,4,4,4,4,5,4,5,5,5,5,6,5,6,6,6,6,
%T A010764 7,6,7,7,7,7,8,7,8,8,8,8,9,8,9,9,9,9,10,9,10,10,10,10,11,10,11,11,11,11,
%U A010764 12,11,12,12,12,12,13,12,13,13,13,13,14,13,14,14,14,14,15,14,15,15
%N A010764 [ n/2 ]%[ n/3 ].
%F A010764 G.f.: x^6(1+x-x^2-x^3+x^4+2x^5-2x^7)/((1-x^2)(1-x^3)).
%p A010764 [ seq(floor(n/2) mod floor(n/3), n=3..100) ];
%o A010764 (PARI) a(n)=if(n<3,0,(n\2)%(n\3))
%Y A010764 a(n)=A008615(n), n>8.
%Y A010764 Sequence in context: A054725 A064415 A086833 this_sequence A029383 A070098 A029233
%Y A010764 Adjacent sequences: A010761 A010762 A010763 this_sequence A010765 A010766 A010767
%K A010764 nonn
%O A010764 3,8
%A A010764 njas, Simon Plouffe (plouffe(AT)math.uqam.ca)
%I A029383
%S A029383 1,0,0,0,0,1,1,0,1,1,1,1,1,1,2,2,2,2,3,2,3,3,3,4,5,4,5,
%T A029383 5,5,6,7,6,8,8,8,9,10,9,11,11,12,13,14,13,15,16,16,17,19,
%U A029383 18,21,21,21,23,25,24,27,27,28,30,32,31,34,35,36,38,40
%N A029383 Expansion of 1/((1-x^5)(1-x^6)(1-x^8)(1-x^9)).
%Y A029383 Sequence in context: A064415 A086833 A010764 this_sequence A070098 A029233 A051888
%Y A029383 Adjacent sequences: A029380 A029381 A029382 this_sequence A029384 A029385 A029386
%K A029383 nonn
%O A029383 0,15
%A A029383 njas
%I A070098
%S A070098 0,0,1,0,1,1,1,1,2,2,2,2,3,2,3,3,4,3,4,4,4,4,5,4,5,5,6,5,6,6,6,6,7,7,7,
%T A070098 7,8,7,8,8,8,8,9,9,9,9,10,9,10,10,11,10,11,11,11,11,12,12,12,12,13,12,
%U A070098 13,13,13,13,14,14,14,14,15,14,15,15,16,15
%N A070098 Number of integer triangles with perimeter n which are acute and isosceles.
%C A070098 a(n)=A070093(n)-A024154(n); a(n)=A059169(n)-A070106(n).
%H A070098 R. Zumkeller, Integer-sided triangles
%e A070098 For n=9 there are A005044(9)=3 integer triangles: [1,4,4], [2,3,4], and [3,3,3]; both isosceles are also acute.
%Y A070098 Cf. A070080, A070081, A070082, A059169, A070099, A070100, A070124.
%Y A070098 Sequence in context: A086833 A010764 A029383 this_sequence A029233 A051888 A088019
%Y A070098 Adjacent sequences: A070095 A070096 A070097 this_sequence A070099 A070100 A070101
%K A070098 nonn
%O A070098 1,9
%A A070098 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 05 2002
%I A029233
%S A029233 1,0,1,0,1,0,1,1,1,1,2,2,2,2,3,2,3,3,4,3,5,5,6,5,7,6,7,
%T A029233 7,9,8,10,10,12,11,13,13,14,14,16,16,18,18,21,20,23,22,
%U A029233 25,24,27,27,30,30,33,33,36,36,39,39,42,42,46,46,50,50
%N A029233 Expansion of 1/((1-x^2)(1-x^7)(1-x^10)(1-x^11)).
%Y A029233 Sequence in context: A010764 A029383 A070098 this_sequence A051888 A088019 A029348
%Y A029233 Adjacent sequences: A029230 A029231 A029232 this_sequence A029234 A029235 A029236
%K A029233 nonn
%O A029233 0,11
%A A029233 njas
%I A051888
%S A051888 2,2,2,2,3,2,3,3,7,3,3,5,2,3,13,7,31,5,2,7,17,67,41,3,13,3,43,17,97,7,29,
%T A051888 109,3,71,5,2,7,41,3,59,3,11,29,7,107,67,79,3,743,149,163,2,211,2,19,
%U A051888 71,73,23,37,113,149,67,41,617,107,37,107,283,113,19,239,107,73,97,5
%N A051888 a(n) = smallest prime p such that p*n! + 1 is prime.
%F A051888 Analogous to or subset of A051686; generalization of A005384
%t A051888 Do[k = 1; While[ !PrimeQ[ Prime[k]*n! + 1], k++ ]; Print[ Prime[k]], {n, 1, 75} ]
%Y A051888 A005384, A051686, A051886, A051887.
%Y A051888 Sequence in context: A029383 A070098 A029233 this_sequence A088019 A029348 A070093
%Y A051888 Adjacent sequences: A051885 A051886 A051887 this_sequence A051889 A051890 A051891
%K A051888 nonn
%O A051888 0,1
%A A051888 Labos E. (labos(AT)ana.sote.hu), Dec 15 1999
%E A051888 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 16 1999
%I A088019
%S A088019 0,1,2,2,2,2,3,2,3,4,4,3,3,2,3,4,4,3,3,2,3,4,4,4,4,4,4,4,4,4,5,4,4,4,4,
%T A088019 5,6,6,6,6,6,5,5,4,4,4,4,4,4,4,5,6,6,7,8,8,8,8,8,7,7,6,6,6,6,6,6,6,7,8,
%U A088019 8,7,7,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,11,12,12,13,14
%N A088019 Number of twin primes between n and 2n (inclusive).
%C A088019 Here a twin prime is counted even if only one member of the twin-prime pair is between n and 2n, inclusive. Note that this sequence is very close to 2*A088018. It appears that a(n) > 0 for all n > 1. However, it has not been proved that there are an infinite number of twin primes.
%H A088019 T. D. Noe, Plot of A088018 for n < 10000
%H A088019 E. W. Weisstein, The World of Mathematics: Twin Primes
%t A088019 pl=Prime[Range[PrimePi[20000]]]; twl={}; Do[If[pl[[i-1]]+2==pl[[i]], twl=Join[twl, {pl[[i-1]], pl[[i]]}]], {i, 2, Length[pl]}]; twl=Union[twl]; i1=1; i2=1; nMin=(twl[[1]]-1)/2; nMax=(twl[[ -1]]+1)/2; Join[Table[0, {nMin-1}], Table[While[twl[[i1]]
%Y A088019 Cf. A035250 (number of primes between n and 2n), A088018 (number of twin-prime pairs between n and 2n).
%Y A088019 Sequence in context: A070098 A029233 A051888 this_sequence A029348 A070093 A058744
%Y A088019 Adjacent sequences: A088016 A088017 A088018 this_sequence A088020 A088021 A088022
%K A088019 easy,nonn
%O A088019 1,3
%A A088019 T. D. Noe (noe(AT)sspectra.com), Sep 18 2003
%I A029348
%S A029348 1,0,0,0,1,0,1,1,1,1,1,1,2,2,2,2,3,2,4,3,4,4,5,4,6,6,6,
%T A029348 7,8,7,9,9,10,10,12,11,14,13,14,15,17,16,19,19,20,21,23,
%U A029348 22,26,26,27,28,31,30,34,34,36,37,40,39,44,44,46,48,51
%N A029348 Expansion of 1/((1-x^4)(1-x^6)(1-x^7)(1-x^9)).
%Y A029348 Sequence in context: A029233 A051888 A088019 this_sequence A070093 A058744 A081743
%Y A029348 Adjacent sequences: A029345 A029346 A029347 this_sequence A029349 A029350 A029351
%K A029348 nonn
%O A029348 0,13
%A A029348 njas
%I A070093
%S A070093 0,0,1,0,1,1,1,1,2,2,2,2,3,2,4,3,5,4,5,5,5,6,6,6,7,7,9,8,10,9,10,10,11,
%T A070093 12,12,12,14,13,16,14,17,16,17,18,18,20,20,20,22,22,24,23,25,26,26,27,
%U A070093 28,30,30,29,32,31,35,33,36,36,38,39,40,40
%N A070093 Number of acute integer triangles with perimeter n.
%C A070093 An integer triangle [A070080(k)<=A070081(k)<=A070082(k)] is acute iff A070085(k)>0;
%C A070093 a(n) = A005044(n) - A070101(n) - A024155(n);
%C A070093 a(n) = A042154(n) + A070098(n).
%H A070093 E. W. Weisstein, Acute Triangle.
%H A070093 R. Zumkeller, Integer-sided triangles
%e A070093 For n=9 there are A005044(9)=3 integer triangles: [1,4,4], [2,3,4], and [3,3,3]; two of them are acute, as 2^2+3^2<16=4^2, therefore a(9)=2.
%Y A070093 Cf. A070094, A070095, A070118.
%Y A070093 Sequence in context: A051888 A088019 A029348 this_sequence A058744 A081743 A118377
%Y A070093 Adjacent sequences: A070090 A070091 A070092 this_sequence A070094 A070095 A070096
%K A070093 nonn
%O A070093 1,9
%A A070093 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 05 2002
%I A058744
%S A058744 1,0,1,0,2,2,2,2,3,2,4,4,7,4,10,8,11,10,14,14,21,18,25,22,33
%N A058744 McKay-Thompson series of class 70A for Monster.
%D A058744 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No.13, 5175-5193 (1994).
%Y A058744 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%Y A058744 Sequence in context: A088019 A029348 A070093 this_sequence A081743 A118377 A023516
%Y A058744 Adjacent sequences: A058741 A058742 A058743 this_sequence A058745 A058746 A058747
%K A058744 nonn
%O A058744 -1,5
%A A058744 njas, Nov 27, 2000
%I A081743
%S A081743 1,2,2,2,2,3,3,2,2,3,3,3,3,4,4,2,2,3,3,3,3,4,4,3,3,4,4,4,4,5,5,2,2,3,3,
%T A081743 3,3,4,4,3,3,4,4,4,4,5,5,3,3,4,4,4,4,5,5,4,4,5,5,5,5,6,6,2,2,3,3,3,3,4,
%U A081743 4,3,3,4,4,4,4,5,5,3,3,4,4,4,4,5,5,4,4,5,5,5,5,6,6,3,3,4,4,4,4,5,5,4,4
%N A081743 a(1)=1 then a(n)=a(n/2^k)+1 if n is even and 2^k is the largest power of 2 dividing n, a(n)=a(n-1) otherwise.
%p A081743 a(n)=if(n<2,1,if(n%2,a(n-1),a(n/2^valuation(n,2))+1))
%Y A081743 Sequence in context: A029348 A070093 A058744 this_sequence A118377 A023516 A096198
%Y A081743 Adjacent sequences: A081740 A081741 A081742 this_sequence A081744 A081745 A081746
%K A081743 nonn
%O A081743 1,2
%A A081743 Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 07 2003
%I A118377
%S A118377 2,2,2,2,3,3,2,2,13,3,13,2,3,11,7,37,151,11,113,2,5,2,401,73,7,109,3,7,
%T A118377 101,2,11,109,5,277,11,7,31,89,191,31,11,2713,11,13,73,461
%N A118377 Least prime p such that p(n)#*p#-1 is prime.
%e A118377 2*2-1=3 prime 2=p(1)# so a(1)=2
%e A118377 2*3*2-1=11 prime 2*3=p(2)# so a(2)=2
%e A118377 2*3*5*2-1=59 prime 2*3*5=p(3)# so a(3)=2
%e A118377 2*3*5*7*2-1=419 prime 2*3*5*7=p(4)# so a(4)=2
%e A118377 2*3*5*7*11*2*3-1=13859 prime 2*3*5*7*11=p(5)# so a(5)=3
%Y A118377 Sequence in context: A070093 A058744 A081743 this_sequence A023516 A096198 A103183
%Y A118377 Adjacent sequences: A118374 A118375 A118376 this_sequence A118378 A118379 A118380
%K A118377 nonn
%O A118377 1,1
%A A118377 Pierre CAMI (pierrecami(AT)tele2.fr), May 15 2006
%I A023516
%S A023516 0,1,2,2,2,2,3,3,2,3,2,3,2,2,3,4,3,3,3,3,2,3,3,3,3,3,3,4,3,2,4,2,3,
%T A023516 2,4,3,3,4,3,4,4,3,3,3,3,3,3,3,3,4,2,3,3,4,4,4,4,4,3,4,2,3,4,2,4,4,
%U A023516 3,3,3,3,3,3,3,4,3,3,4,3,2,4
%N A023516 Number of distinct prime divisors of prime(n)*prime(n-1) - 1.
%Y A023516 Sequence in context: A058744 A081743 A118377 this_sequence A096198 A103183 A115263
%Y A023516 Adjacent sequences: A023513 A023514 A023515 this_sequence A023517 A023518 A023519
%K A023516 nonn
%O A023516 1,3
%A A023516 Clark Kimberling (ck6(AT)evansville.edu)
%I A096198
%S A096198 0,1,1,2,2,2,2,3,3,2,3,3,4,3,3,3,4,4,4,4,3,3,4,5,4,5,4,3,3,4,5,5,5,5,4,
%T A096198 3,4,4,5,5,6,5,5,4,4,4,5,5,5,6,6,5,5,5,4,4,5,6,5,6,6,6,5,6,5,4,4,5,6,6,
%U A096198 6,6,6,6,6,6,5,4,4,5,6,6,7,6,6,6,7,6,6,5,4,4,5,6,6,7,7,6,6,7,7,6,6,5,4
%N A096198 Triangle read by rows: T(m,n)=A029837(m)+A029837(n), where (m,n)=(1,1); (2,1), (1,2); (3,1), (2,2), (1,3); ...
%C A096198 A029837(n) is the smallest k such that 2^k>=n. T(m,n) is the solution to the following simple problem. What is the minimum number of cuts needed to divide a sheet of paper whose sides are in the ratio m:n into mn square pieces of equal size? (A single cut means either cutting one rectangle into two smaller rectangles or placing two or more sheets on top of one another and cutting through the lot in one go.)
%e A096198 Array begins
%e A096198 0
%e A096198 1 1
%e A096198 2 2 2
%e A096198 2 3 3 2
%e A096198 3 3 4 3 3
%Y A096198 Cf. A029837.
%Y A096198 Sequence in context: A081743 A118377 A023516 this_sequence A103183 A115263 A055894
%Y A096198 Adjacent sequences: A096195 A096196 A096197 this_sequence A096199 A096200 A096201
%K A096198 easy,nonn,tabl
%O A096198 1,4
%A A096198 Paul Boddington (psb(AT)maths.warwick.ac.uk), Jul 26 2004
%I A103183
%S A103183 0,1,1,2,2,2,2,3,3,2,3,3,4,3,4,4,3,4,4,3,4,4,5,5,5,5,3,4,5,4,5,6,4,
%T A103183 5,6,6,6,4,5,6,5,6,5,6,4,5,6,5,7,6,7,7,7,6,7,4,5,6,7,5,6,7,6,7,7,8,
%U A103183 5,6,7,8,8,8,8,5,6,7,8,6,7,8,8,6,7,8,5,6,7,8,6,7,9,8,9,9,7,9,8,9,7
%N A103183 Values of k corresponding to terms in A103182.
%Y A103183 Sequence in context: A118377 A023516 A096198 this_sequence A115263 A055894 A120425
%Y A103183 Adjacent sequences: A103180 A103181 A103182 this_sequence A103184 A103185 A103186
%K A103183 nonn
%O A103183 1,4
%A A103183 Eric Angelini (keynews.tv(AT)skynet.be), Mar 18 2005
%E A103183 More terms from Kerry Mitchell (lkmitch(AT)att.net), Mar 09 2005
%I A115263
%S A115263 1,1,1,2,2,2,2,3,3,2,3,4,6,4,3,3,5,7,7,5,3,4,6,10,10,10,6,4,4,7,11,13,
%T A115263 13,11,7,4,5,8,14,16,19,16,14,8,5,5,9,15,19,22,22,19,15,9,5,6,10,18,22,
%U A115263 28,28,28,22,18,10,6
%N A115263 Correlation triangle for floor((n+2)/2).
%C A115263 Row sums are A096338. Diagonal sums are A115264. T(2n,n) is A005993. T(2n,n)-T(2n,n+1) is floor((n+2)/2)(1+(-1)^n)/2 (aerated n+1).
%F A115263 G.f.: (1+x)(1+xy)/((1-x^2)^2*(1-x^2*y^2)^2*(1-x^2*y)); Number triangle T(n, k)=sum{j=0..n, [j<=k]*floor((k-j+2)/2)*[j<=n-k]*floor((n-k-j+2)/2)}.
%e A115263 Triangle begins
%e A115263 1;
%e A115263 1,1;
%e A115263 2,2,2;
%e A115263 2,3,3,2;
%e A115263 3,4,6,4,3;
%e A115263 3,5,7,7,5,3;
%Y A115263 Sequence in context: A023516 A096198 A103183 this_sequence A055894 A120425 A104186
%Y A115263 Adjacent sequences: A115260 A115261 A115262 this_sequence A115264 A115265 A115266
%K A115263 easy,nonn,tabl
%O A115263 0,4
%A A115263 Paul Barry (pbarry(AT)wit.ie), Jan 18 2006
%I A055894
%S A055894 1,1,1,2,2,2,2,3,3,2,3,4,8,4,3,2,5,10,10,5,2,4,6,18,22,18,6,4,2,7,21,
%T A055894 35,35,21,7,2,4,8,32,56,78,56,32,8,4,3,9,36,87,126,126,87,36,9,3,4,10,
%U A055894 50,120,220,254,220,120,50,10,4,2,11,55,165,330,462,462,330,165,55,11
%N A055894 Inverse Moebius transform of Pascal's triangle A007318.
%H A055894 N. J. A. Sloane, Transforms
%H A055894 Index entries for triangles and arrays related to Pascal's triangle
%e A055894 1; 1,1; 2,2,2; 2,3,3,2; 3,4,8,4,3; ...
%Y A055894 Row sums give A055895.
%Y A055894 Sequence in context: A096198 A103183 A115263 this_sequence A120425 A104186 A092363
%Y A055894 Adjacent sequences: A055891 A055892 A055893 this_sequence A055895 A055896 A055897
%K A055894 nonn,tabl
%O A055894 0,4
%A A055894 Christian G. Bower (bowerc(AT)usa.net), Jun 09 2000
%I A120425
%S A120425 1,1,2,2,2,2,3,3,3,2,2,2,4,4,4,4,3,3,3,3,5,5,5,5,5,4,4,4,4,4,6,6,6,6,6,
%T A120425 6,4,4,4,4,5,5,7,7,7,7,7,7,7,5,4,4,4,4,4,4,8,8,8,8,8,8,8,8,6,6,6,6,6,6,
%U A120425 6,6,9,9,9,9,9,9,9,9,9,5,5,5,7,7,7,7,7,7,10,10,10,10,10,10,10,10,10,10
%N A120425 a(n) = maximum value among all k where 1<=k<=n of GCD(k,ceiling(n/k)).
%e A120425 For n = 10, we have the pairs {k,ceiling(n/k)} of {1,10},{2,5},{3,4},{4,3},{5,2},{6,2},{7,2},{8,2},{9,2},{10,1}. The GCD's of these 10 pairs are 1,1,1,1,1,2,1,2,1,1. Of these, 2 is the largest. So a(10) = 2.
%t A120425 Table[Max[Table[GCD[k, Ceiling[n/k]], {k, 1, n}]], {n, 1, 100}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Jul 22 2006
%Y A120425 Sequence in context: A103183 A115263 A055894 this_sequence A104186 A092363 A053384
%Y A120425 Adjacent sequences: A120422 A120423 A120424 this_sequence A120426 A120427 A120428
%K A120425 nonn
%O A120425 1,3
%A A120425 Leroy Quet (qq-quet(AT)mindspring.com), Jul 12 2006
%E A120425 More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Jul 22 2006
%I A104186
%S A104186 2,2,2,2,3,3,3,2,2,4,2,1,3,3,2
%N A104186 Number of distinct prime divisors of numbers from four consecutive concatenated odd numbers.
%C A104186 Interestingly, 23252729 is prime.
%e A104186 The number of distinct prime factors of 1357 is 2 - the first term in the sequence.
%e A104186 The number of distinct prime factors of 3579 is 2 - the second term in the sequence.
%e A104186 The number of distinct prime factors of 57911 is 2 - the third term in the sequence.
%Y A104186 Sequence in context: A115263 A055894 A120425 this_sequence A092363 A053384 A069624
%Y A104186 Adjacent sequences: A104183 A104184 A104185 this_sequence A104187 A104188 A104189
%K A104186 nonn,base
%O A104186 0,1
%A A104186 Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Apr 16 2005
%I A092363
%S A092363 2,2,2,2,3,3,3,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,2,2,3,3,3,5,5,5,5,5,5,5,2,
%T A092363 5,5,5,5,41,42,43,44,45,46,47,2,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,
%U A092363 6,6,6,6,6,6,6,4,4,4,4,4,4,4,4,4,4,4,87,88,89,90,91,92,93,94,95,96,97
%N A092363 n^(1/a(n)) is the closest to an integer on 2..n with a(n) minimal.
%C A092363 The sequence is conjectured to tend to n, as n^(1/n)->1. Is the density of non-n entries 0?
%e A092363 5^(1/2)= 2.236067977499789696409173668
%e A092363 5^(1/3)= 1.709975946676696989353108872
%e A092363 5^(1/4)= 1.495348781221220541911898994
%e A092363 5^(1/5)= 1.379729661461214832390063464
%e A092363 5^(1/2) is closest to an integer, so a(5)=2.
%o A092363 (PARI) { for (i=2,100, xj=1;xm=0.5; for (j=2,i, x=i^(1/j)*1.0; xf=x-floor(x); if (xf
%Y A092363 Sequence in context: A055894 A120425 A104186 this_sequence A053384 A069624 A092139
%Y A092363 Adjacent sequences: A092360 A092361 A092362 this_sequence A092364 A092365 A092366
%K A092363 nonn
%O A092363 2,1
%A A092363 Jon Perry (perry(AT)globalnet.co.uk), Mar 19 2004
%I A053384
%S A053384 2,2,2,2,3,3,3,3,2,2,2,2,4,4,4,4,2,2,2,2,3,3,3,3,2,2,2,2,5,5,5,5,2,2,2,
%T A053384 2,3,3,3,3,2,2,2,2,4,4,4,4,2,2,2,2,3,3,3,3,2,2,2,2,6,6,6,6,2,2,2,2,3,3,
%U A053384 3,3,2,2,2,2,4,4,4,4,2,2,2,2,3,3,3,3,2,2,2,2,5,5,5,5,2,2,2,2,3,3,3,3,2
%N A053384 A053398(4, n).
%Y A053384 Sequence in context: A120425 A104186 A092363 this_sequence A069624 A092139 A084558
%Y A053384 Adjacent sequences: A053381 A053382 A053383 this_sequence A053385 A053386 A053387
%K A053384 nonn
%O A053384 1,1
%A A053384 David W. Wilson (davidwwilson(AT)comcast.net)
%I A069624
%S A069624 1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,
%T A069624 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%U A069624 4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5
%N A069624 Number of different values taken by the integer part of n^(1/k) for all k >= 1.
%e A069624 a(28) = 4, floor[28^(1/2)] = 5, floor[28^1/3] = 3, floor[28^1/4] = 2, floor [28^1/5] = 1= floor[28^1/k, (k > 5)], etc. The distinct values are 1,2,3 and 5.
%t A069624 f[n_] := Block[{k = 1}, While[ Floor[n^(1/k)] != 1, k++ ]; k]; Table[ Length[ Union[ Table[ Floor[n^(1/k)], {k, 2, f[n]+1}]]], {n, 1, 105}]
%Y A069624 Cf. A071913.
%Y A069624 Sequence in context: A104186 A092363 A053384 this_sequence A092139 A084558 A066339
%Y A069624 Adjacent sequences: A069621 A069622 A069623 this_sequence A069625 A069626 A069627
%K A069624 nonn
%O A069624 1,4
%A A069624 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 27 2002
%E A069624 Edited by njas and Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 14 2002
%I A092139
%S A092139 0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4
%N A092139 Same as A084558.
%Y A092139 Sequence in context: A092363 A053384 A069624 this_sequence A084558 A066339 A052375
%Y A092139 Adjacent sequences: A092136 A092137 A092138 this_sequence A092140 A092141 A092142
%K A092139 dead
%O A092139 0,3
%I A084558
%S A084558 0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,
%T A084558 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%U A084558 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4
%N A084558 a(n) = largest m such that n >= m!.
%C A084558 The number of significant digits in n's factorial expansion (A007623).
%C A084558 After zero, which occurs once, each n occurs A001563(n) times.
%D A084558 F. Smarandache, "f-Inferior and f-Superior Functions - Generalization of Floor Functions", Arizona State University, Special Collections.
%D A084558 Yi Yuan and Zhang Wenpeng, On the Mean Value of the Analogue of Smarandache Function, Smarandache Notions J., Vol. 15 (to appear).
%H A084558 Yi Yuan and Zhang Wenpeng, On the Mean Value of the Analogue of Smarandache Function.
%e A084558 a(4) = 2 because 2!<=4<3!.
%Y A084558 A dual to A090529.
%Y A084558 Cf. A084555-A084557.
%Y A084558 Sequence in context: A053384 A069624 A092139 this_sequence A066339 A052375 A074279
%Y A084558 Adjacent sequences: A084555 A084556 A084557 this_sequence A084559 A084560 A084561
%K A084558 nonn
%O A084558 0,3
%A A084558 Antti Karttunen (HisFirstname.HisSurname(AT)iki.fi) Jun 23 2003
%I A066339
%S A066339 0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,
%T A066339 4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,
%U A066339 8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11
%N A066339 Number of primes p of the form 4m+1 with p <= n.
%C A066339 Asymptotic expansion: a(n) ~ pi(n)/2 ~ n/(2log(n)) (pi(n) is in sequence A000720).
%H A066339 R. Breusch, An Asymptotic Formula for Primes Of The Form 4n+1
%t A066339 Table[ Length[ Select[ Union[ Table[ Prime[ PrimePi[i]], {i, 2, n}]], Mod[ #, 4] == 1 & ]], {n, 2, 100} ]
%o A066339 (PARI) for(n=1,200,print1(sum(i=1,n,if((i*isprime(i)-1)%4,0,1)),","))
%Y A066339 Cf. A000720.
%Y A066339 Sequence in context: A069624 A092139 A084558 this_sequence A052375 A074279 A072750
%Y A066339 Adjacent sequences: A066336 A066337 A066338 this_sequence A066340 A066341 A066342
%K A066339 nonn
%O A066339 1,13
%A A066339 Sharon Sela (sharonsela(AT)hotmail.com), Jan 01 2002
%E A066339 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 03 2002
%I A052375
%S A052375 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,
%T A052375 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,
%U A052375 4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8
%N A052375 Number of occurrences of least frequent digit in decimal expansion of Pi.
%e A052375 a(50)=1 because 0 appears once in the first 50 digits of pi, and other digits appear more frequently
%Y A052375 Cf. A000796.
%Y A052375 Sequence in context: A092139 A084558 A066339 this_sequence A074279 A072750 A029835
%Y A052375 Adjacent sequences: A052372 A052373 A052374 this_sequence A052376 A052377 A052378
%K A052375 nonn,base
%O A052375 1,51
%A A052375 Henry Bottomley (se16(AT)btinternet.com), Mar 08 2000
%I A074279
%S A074279 1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,
%T A074279 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5
%N A074279 n appears n^2 times.
%C A074279 Since the last occurrence of n comes one before the first occurrence of n+1, and the former is at SUM[i=0..n](i^2) = A000330(n), we have a(A000330(n)) = a(n*(n+1)*(2n+1)/6) = n, and a(1+A000330(n)) = a(1+(n*(n+1)*(2n+1)/6)) = n+1. The current sequence is, loosely, the inverse function of the square pyramidal sequence. See also: A000330 Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6. A000330 has many alternative formulae, thus yielding many alternative formulae for the current sequence. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Mar 18 2006
%H A074279 Y.-F. S. Petermann, J.-L. Remy and I. Vardi, Discrete derivatives of sequences, Adv. in Appl. Math. 27 (2001), 562-84.
%Y A074279 Cf. A000217, A000330, A006331, A050446, A050447, A000537, A006003, A005900.
%Y A074279 Sequence in context: A084558 A066339 A052375 this_sequence A072750 A029835 A074280
%Y A074279 Adjacent sequences: A074276 A074277 A074278 this_sequence A074280 A074281 A074282
%K A074279 nonn
%O A074279 0,2
%A A074279 Jon Perry (perry(AT)globalnet.co.uk), Sep 21 2002
%I A072750
%S A072750 0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,
%T A072750 5,5,5,5,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,9,
%U A072750 9,9,9,9,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,12
%N A072750 Counting factor 7 in square-free numbers <=n.
%e A072750 The first 10 square-free numbers are: 1, 2, 3, 5, 6=2*3, 7, 10=2*5, 11, 13, and 14=2*7
%e A072750 7 and 14 are divisible by 7, therefore a(10)=2.
%Y A072750 Cf. A005117, A072747, A072748, A072749, A072751.
%Y A072750 Sequence in context: A066339 A052375 A074279 this_sequence A029835 A074280 A000523
%Y A072750 Adjacent sequences: A072747 A072748 A072749 this_sequence A072751 A072752 A072753
%K A072750 nonn
%O A072750 1,10
%A A072750 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 08 2002
%I A029835
%S A029835 0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,
%T A029835 4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,
%U A029835 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6
%N A029835 [ log(n-th prime) ].
%Y A029835 Sequence in context: A052375 A074279 A072750 this_sequence A074280 A000523 A072749
%Y A029835 Adjacent sequences: A029832 A029833 A029834 this_sequence A029836 A029837 A029838
%K A029835 nonn,easy
%O A029835 0,5
%A A029835 njas
%I A074280
%S A074280 0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5
%N A074280 Same as A000523.
%Y A074280 Sequence in context: A074279 A072750 A029835 this_sequence A000523 A072749 A066490
%Y A074280 Adjacent sequences: A074277 A074278 A074279 this_sequence A074281 A074282 A074283
%K A074280 dead
%O A074280 1,4
%I A000523
%S A000523 0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,
%T A000523 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,
%U A000523 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6
%N A000523 Log_2(n) rounded down.
%C A000523 Or, n-1 appears 2^(n-1) times. - Jon Perry (perry(AT)globalnet.co.uk), Sep 21 2002
%C A000523 a(n) + 1 = number of bits in binary expansion of n.
%C A000523 Largest power of 2 dividing LCM[1..n]: A007814[A003418(n)].
%C A000523 Log_2(0) = -infinity.
%C A000523 Also max(Omega(k): 1<=k<=n), where Omega(n)=A001222(n), number of prime factors with repetition; see A080613. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Feb 25 2003
%C A000523 a(n+1) = number of digits of n-th number with no 0 in ternary representation = A081604(A032924(n)); A107680(n) = A003462(a(n+1)). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 20 2005
%D A000523 G. H. Hardy, Note on Dr. Vacca's series..., Quart. J. Pure Appl. Math. 43 (1912) 215-216.
%D A000523 D. E. Knuth, The Art of Computer Programming, Vol. 1: Fundamental Algorithms, p. 400.
%H A000523 R. Stephan, Some divide-and-conquer sequences ...
%H A000523 R. Stephan, Table of generating functions
%F A000523 a(n) = if n > 1 then a(floor(n / 2)) + 1 else 0. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Oct 29 2001
%F A000523 G.f.: 1/(1-x) * Sum(k>=1, x^2^k). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 13 2002
%e A000523 a(5)=2 because the binary expansion of 5 (=101) has three bits.
%p A000523 A000523 := n->floor(simplify(log(n)/log(2)));
%p A000523 A000523 := proc(n) local nn,i; if(0 = n) then RETURN(-infinity); fi; nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi; nn := floor(nn/2); od; end;
%o A000523 (PARI) a(n)=if(n<1,0,floor(log(n)/log(2)))
%Y A000523 Cf. A029837. Partial sums: A061168.
%Y A000523 a(n) = A070939(n)-1 for n>=1.
%Y A000523 Sequence in context: A072750 A029835 A074280 this_sequence A072749 A066490 A090973
%Y A000523 Adjacent sequences: A000520 A000521 A000522 this_sequence A000524 A000525 A000526
%K A000523 nonn,easy,nice
%O A000523 1,4
%A A000523 njas
%E A000523 Error in 4th term, pointed out by Joe Keane (jgk(AT)jgk.org), has been corrected.
%E A000523 More terms from Michael Somos, Aug 02, 2002
%I A072749
%S A072749 0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,
%T A072749 6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,11,11,
%U A072749 11,11,12,12,12,12,13,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14
%N A072749 Counting factor 5 in square-free numbers <=n.
%e A072749 The first 10 square-free numbers are: 1, 2, 3, 5, 6=2*3, 7, 10=2*5, 11, 13, and 14=2*7
%e A072749 5 and 10 are divisible by 5, therefore a(10)=2.
%Y A072749 Cf. A005117, A072747, A072748, A072750, A072751.
%Y A072749 Sequence in context: A029835 A074280 A000523 this_sequence A066490 A090973 A076634
%Y A072749 Adjacent sequences: A072746 A072747 A072748 this_sequence A072750 A072751 A072752
%K A072749 nonn
%O A072749 1,7
%A A072749 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 08 2002
%I A066490
%S A066490 0,0,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,
%T A066490 6,6,6,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,
%U A066490 10,10,11,11,11,11,11,11,11,11,12,12,12,12,13,13,13,13,13,13,13,13,13
%N A066490 Number of primes of the form 4m+3 <= n.
%t A066490 Table[ Length[ Select[ Union[ Table[ Prime[ PrimePi[i]], {i, 2, n}]], Mod[ #, 4] == 3 & ]], {n, 2, 100} ]
%o A066490 (PARI) for(n=1,100,print1(sum(i=1,n,if((i*isprime(i)-3)%4,0,1)),","))
%Y A066490 Cf. A066339.
%Y A066490 Sequence in context: A074280 A000523 A072749 this_sequence A090973 A076634 A083277
%Y A066490 Adjacent sequences: A066487 A066488 A066489 this_sequence A066491 A066492 A066493
%K A066490 easy,nonn
%O A066490 1,7
%A A066490 Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 03 2002
%I A090973
%S A090973 2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,
%T A090973 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,5,
%U A090973 5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6
%N A090973 a(n) = ceil((prime(n)/n).
%C A090973 For n > 1, a(n) = A038605(n)+1. - David Wasserman (wasserma(AT)spawar.navy.mil), Feb 23 2006
%e A090973 a(12) = 4 as pi(48) = 15 > 12 > pi(36) = 11.
%Y A090973 Cf. A038606.
%Y A090973 Sequence in context: A000523 A072749 A066490 this_sequence A076634 A083277 A064557
%Y A090973 Adjacent sequences: A090970 A090971 A090972 this_sequence A090974 A090975 A090976
%K A090973 nonn
%O A090973 1,1
%A A090973 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jan 04 2004
%E A090973 More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Feb 23 2006
%I A076634
%S A076634 1,1,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,
%T A076634 5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,
%U A076634 6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7
%N A076634 Coefficient of x^a(n) in (x+1/2)*(x+2/2)*...*(x+n/2) is the largest one.
%Y A076634 Sequence in context: A072749 A066490 A090973 this_sequence A083277 A064557 A064601
%Y A076634 Adjacent sequences: A076631 A076632 A076633 this_sequence A076635 A076636 A076637
%K A076634 nonn
%O A076634 1,3
%A A076634 Benoit Cloitre (abmt(AT)wanadoo.fr), Nov 10 2002
%I A083277
%S A083277 1,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,
%T A083277 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,
%U A083277 8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8
%N A083277 n appears 3n-2 times.
%F A083277 ceil((1+sqrt(24n+1))/6) or floor((7+sqrt(24n-23))/6)
%Y A083277 Sequence in context: A066490 A090973 A076634 this_sequence A064557 A064601 A023967
%Y A083277 Adjacent sequences: A083274 A083275 A083276 this_sequence A083278 A083279 A083280
%K A083277 nonn
%O A083277 1,2
%A A083277 Daniele Parisse (daniele.parisse(AT)m.eads.net), Jun 02 2003
%I A064557
%S A064557 0,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,
%T A064557 5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,
%U A064557 8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9
%N A064557 a(n) = # { p | A064553(k) = p prime and k <= n}.
%C A064557 Primes occur in A064553 in natural order but less dense.
%Y A064557 A000720, A064553.
%Y A064557 Sequence in context: A090973 A076634 A083277 this_sequence A064601 A023967 A090532
%Y A064557 Adjacent sequences: A064554 A064555 A064556 this_sequence A064558 A064559 A064560
%K A064557 nonn
%O A064557 1,3
%A A064557 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Sep 21 2001
%I A064601
%S A064601 0,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,
%T A064601 5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,
%U A064601 8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9
%N A064601 a(n) = # { p | A064558(k) = p prime and k <= n}.
%C A064601 As in A064553 primes occur in A064558 in natural order but are far less dense.
%Y A064601 Cf. A064558, A000720, A064557, A064600.
%Y A064601 Sequence in context: A076634 A083277 A064557 this_sequence A023967 A090532 A003058
%Y A064601 Adjacent sequences: A064598 A064599 A064600 this_sequence A064602 A064603 A064604
%K A064601 nonn
%O A064601 1,3
%A A064601 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Sep 22 2001
%I A023967
%S A023967 0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,
%T A023967 5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,
%U A023967 6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7
%N A023967 First digit after decimal point of 8-th root of n.
%t A023967 Array[ Function[ n, RealDigits[ N[ Power[ n, 1/8 ], 10 ], 10 ]// (#[ [ 1, #[ [ 2 ] ]+1 ] ])& ], 110 ]
%Y A023967 Sequence in context: A083277 A064557 A064601 this_sequence A090532 A003058 A000194
%Y A023967 Adjacent sequences: A023964 A023965 A023966 this_sequence A023968 A023969 A023970
%K A023967 nonn,base
%O A023967 1,5
%A A023967 njas, Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A090532
%S A090532 1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,
%T A090532 6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,
%U A090532 8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9
%N A090532 Let f(n) = n - pi(n). Then a(n) = least number of steps such that f(f(...(n)))=1.
%e A090532 a(10) = 3, 10 ->6 ->3 ->1.
%e A090532 a(100) = 9.
%e A090532 f(100) =100-25 = 75, f(75) = 75-21= 54, f(54) = 54-16 = 38, f(38) = 38-12= 26, f(26) = 26-9 = 17, f(17) = 17-7 = 10, f(10) = 10-4 =6, f(6) = 6-3=3, f(3) = 3-2 =1.
%Y A090532 Cf. A025003.
%Y A090532 Sequence in context: A064557 A064601 A023967 this_sequence A003058 A000194 A097429
%Y A090532 Adjacent sequences: A090529 A090530 A090531 this_sequence A090533 A090534 A090535
%K A090532 nonn
%O A090532 2,3
%A A090532 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Dec 07 2003
%E A090532 Corrected and extended by Sam Handler (sam_5_5_5_0(AT)yahoo.com), Dec 11 2004
%I A003058
%S A003058 1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6
%N A003058 Same as A000194.
%Y A003058 Sequence in context: A064601 A023967 A090532 this_sequence A000194 A097429 A100617
%Y A003058 Adjacent sequences: A003055 A003056 A003057 this_sequence A003059 A003060 A003061
%K A003058 dead
%O A003058 1,3
%I A000194
%S A000194 1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,
%T A000194 6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,
%U A000194 8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10
%N A000194 n appears 2n times; also nearest integer to square root of n.
%D A000194 B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 78, Entry 24.
%D A000194 M. A. Nyblom, Some curious sequences ..., Am. Math. Monthly 109 (#6, 200), 559-564.
%H A000194 M. Somos, Sequences used for indexing triangular or square arrays
%F A000194 G.f.: f(x^2, x^6)*x/(1-x) where f(a, b) is Ramanujan's theta function.
%F A000194 a(n)=a(n-2*a(n-a(n-1)))+1. - Benoit Cloitre (abmt(AT)wanadoo.fr), Oct 27 2002
%F A000194 a(n+1)=a(n)+A005369(n).
%F A000194 a(n)=floor((1/2)*(1 + sqrt(4*n - 3))). - Zak Seidov, Jan 18 2006
%p A000194 Digits := 100; f := n->round(evalf(sqrt(n))); [ seq(f(n), n=1..100) ];
%o A000194 (PARI) a(n)=if(n<0,0,ceil(sqrtint(4*n)/2)) - Michael Somos Feb 11 2004
%Y A000194 Partial sums of A005369.
%Y A000194 A000037(n) - n.
%Y A000194 Sequence in context: A023967 A090532 A003058 this_sequence A097429 A100617 A076471
%Y A000194 Adjacent sequences: A000191 A000192 A000193 this_sequence A000195 A000196 A000197
%K A000194 nonn,easy,nice
%O A000194 1,3
%A A000194 njas
%E A000194 Additional comments from Michael Somos, May 31, 2000.
%I A097429
%S A097429 0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,
%T A097429 6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,10,10,10,
%U A097429 10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12
%N A097429 Integer part of the radii of circles with prime areas.
%F A097429 Radius = floor(sqrt(Area/Pi)).
%e A097429 If A = 5. floor(sqrt(5/Pi)) = 1, the third entry.
%o A097429 (PARI) f(n) = forprime(x=1,n,print1(floor(sqrt(x/Pi))","))
%Y A097429 Sequence in context: A090532 A003058 A000194 this_sequence A100617 A076471 A111656
%Y A097429 Adjacent sequences: A097426 A097427 A097428 this_sequence A097430 A097431 A097432
%K A097429 nonn
%O A097429 2,6
%A A097429 Cino Hilliard (hillcino368(AT)hotmail.com), Aug 22 2004
%I A100617
%S A100617 1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,
%T A100617 6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,
%U A100617 9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11
%N A100617 There are n people in a room. First half (i.e. [n/2]) of them leave, then 1/3 (i.e. floor of 1/3) of those remaining leave, then 1/4, then 1/5, etc.; sequence gives number who remain at the end.
%D A100617 V. Brun, Un proc\'{e}d\'{e} qui ressemble au crible d'Eratostene, Analele Stiintifice Univ. "Al. I. Cuza", Iasi, Romania, Sect. Ia Matematica, 1965, vol. 11B, pp. 47-53.
%F A100617 a(n) = k for Fl(k) <= n < Fl(k+1), where Fl(i) = A000960(i).
%e A100617 7 -> 7 - [7/2] = 7 - 3 = 4 -> 4 - [4/3] = 4 - 1 = 3 -> 3 - [3/4] = 3 - 0 = 3, which is now fixed, so a(7) = 3.
%p A100617 f:=proc(n) local i,j,k; k:=n; for i from 2 to 10000 do j := floor(k/i); if j < 1 then break; fi; k := k-j; od; k; end;
%Y A100617 Cf. A000960, A100618.
%Y A100617 Sequence in context: A003058 A000194 A097429 this_sequence A076471 A111656 A025423
%Y A100617 Adjacent sequences: A100614 A100615 A100616 this_sequence A100618 A100619 A100620
%K A100617 nonn,nice
%O A100617 1,3
%A A100617 njas, Dec 03 2004
%I A076471
%S A076471 0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,
%T A076471 7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,
%U A076471 10,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13
%N A076471 Number of pairs (p,q) of successive primes with p+q<=n.
%C A076471 A056172(n)-1 <= a(n) <= A056172(n);
%C A076471 a(n) = A076472(n) + A076473(n).
%e A076471 Pairs (p,q) of successive primes with p+q<=27: {(2,3), (3,5), (5,7), (7,11), (11,13)}, hence a(27)=5.
%Y A076471 Cf. A000720.
%Y A076471 Sequence in context: A000194 A097429 A100617 this_sequence A111656 A025423 A087233
%Y A076471 Adjacent sequences: A076468 A076469 A076470 this_sequence A076472 A076473 A076474
%K A076471 nonn
%O A076471 1,8
%A A076471 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Oct 14 2002
%I A111656
%S A111656 2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,11,11,11,11,11,
%T A111656 11,11,11,11,11,13,13,13,13,13,13,13,13,13,13,13,13,17,17,17,17,17,17,
%U A111656 17,17,17,17,17,17,17,17,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19
%N A111656 nth prime appears nth composite number times.
%D A111656 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
%D A111656 J. B. Rosser and L. Schoenfield, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6: 64-94 (1962).
%D A111656 L. Panaitopol, Some Properties of the Series of Composed Numbers, J. Inequalities in Pure and Applied Mathematics. 2(2): Article 38, 2000.
%H A111656 Eric W. Weisstein, Composite Number."
%H A111656 Eric W. Weisstein, Prime Number."
%F A111656 A000040(n) appears A002808(n) times.
%Y A111656 Cf. A000040, A002808, A111653, A111654, A111655, A111657.
%Y A111656 Sequence in context: A097429 A100617 A076471 this_sequence A025423 A087233 A104147
%Y A111656 Adjacent sequences: A111653 A111654 A111655 this_sequence A111657 A111658 A111659
%K A111656 easy,nonn
%O A111656 1,1
%A A111656 Jonathan Vos Post (jvospost2(AT)yahoo.com), Aug 12 2005
%I A025423
%S A025423 1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,3,5,5,5,5,6,6,7,6,6,8,8,7,8,10,9,9,10,9,
%T A025423 11,10,11,13,14,11,13,14,13,14,14,16,17,15,14,18,18,16,19,21,23,20,21,20,
%U A025423 23,22,19,26,26,23,25,26,26,27,27,28,34,29,28,31,33,30,32,36,35,36,34,36
%N A025423 Number of partitions of n into 8 squares.
%Y A025423 Sequence in context: A100617 A076471 A111656 this_sequence A087233 A104147 A052146
%Y A025423 Adjacent sequences: A025420 A025421 A025422 this_sequence A025424 A025425 A025426
%K A025423 nonn
%O A025423 0,5
%A A025423 David W. Wilson (davidwwilson(AT)comcast.net)
%I A087233
%S A087233 1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,
%T A087233 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,
%U A087233 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6
%N A087233 a(n)=floor[sigma[A002110(n)]/A002110(n)]; integer quotient of divisor-sum of primorial numbers and primorials.
%t A087233 q[x_] := Apply[Times, Table[Prime[w], {w, 1, x}]]; q[0]=1; Table[Floor[DivisorSigma[1, a=q[u]]/q[u]//N], {u, 1, 300}]
%Y A087233 Cf. A000203, A002110.
%Y A087233 Sequence in context: A076471 A111656 A025423 this_sequence A104147 A052146 A097882
%Y A087233 Adjacent sequences: A087230 A087231 A087232 this_sequence A087234 A087235 A087236
%K A087233 nonn
%O A087233 1,2
%A A087233 Labos E. (labos(AT)ana.sote.hu), Sep 01 2003
%I A104147
%S A104147 2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,
%T A104147 6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,
%U A104147 8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9
%N A104147 Number of cubes <= n-th prime.
%Y A104147 Sequence in context: A111656 A025423 A087233 this_sequence A052146 A097882 A108955
%Y A104147 Adjacent sequences: A104144 A104145 A104146 this_sequence A104148 A104149 A104150
%K A104147 easy,nonn
%O A104147 0,1
%A A104147 Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Mar 08 2005
%I A052146
%S A052146 0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,
%T A052146 6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,
%U A052146 9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11
%N A052146 floor((sqrt(1+8*n)-3)/2).
%D A052146 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.2(d).
%Y A052146 Sequence in context: A025423 A087233 A104147 this_sequence A097882 A108955 A108956
%Y A052146 Adjacent sequences: A052143 A052144 A052145 this_sequence A052147 A052148 A052149
%K A052146 nonn
%O A052146 1,6
%A A052146 njas, Jan 23 2000
%I A097882
%S A097882 0,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,
%T A097882 8,8,8,9,9,9,9,9,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,13,
%U A097882 13,12,13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,15,15,16
%N A097882 Floor(n^2/prime(n)).
%C A097882 a(n) = floor(A000290(n)/A000040(n));
%C A097882 A000290(n) = a(n)*A000040(n)) + A069547(n).
%Y A097882 Cf. A001221.
%Y A097882 Sequence in context: A087233 A104147 A052146 this_sequence A108955 A108956 A108037
%Y A097882 Adjacent sequences: A097879 A097880 A097881 this_sequence A097883 A097884 A097885
%K A097882 nonn
%O A097882 1,4
%A A097882 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Sep 02 2004
%I A108955
%S A108955 1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,8,8,8,8,8,9,
%T A108955 9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,12,12,13,13,13,13,
%U A108955 13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17
%N A108955 Floor(Li(2n) - Li(n)).
%F A108955 Li(x) = Int(t=2, x, dt/log(t)) = Logarithmic Integral.
%o A108955 (PARI) L(n) = for(x=2,n,y=Li(2*x)-Li(x);print1(floor(y)",")) Li(x) = \ Logarithmic integral { -eint1(log(1/x)) }
%Y A108955 Sequence in context: A104147 A052146 A097882 this_sequence A108956 A108037 A055679
%Y A108955 Adjacent sequences: A108952 A108953 A108954 this_sequence A108956 A108957 A108958
%K A108955 easy,nonn
%O A108955 2,2
%A A108955 Cino Hilliard (hillcino368(AT)hotmail.com), Jul 22 2005
%I A108956
%S A108956 1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,8,8,8,8,8,9,
%T A108956 9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,12,12,12,12,12,12,13,13,13,13,
%U A108956 13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17
%N A108956 Floor(R(2n) - R(n)).
%F A108956 R(x) = Riemann Pi(x) approximation
%o A108956 (PARI) p(n) = for(x=2,n,y=R(2*x)-R(x);print1(floor(y)",")) R(x) = \ Riemann approx of Pi(x) { local(j); round(sum(j=1,200,moebius(j)*Li(x^(1/j))/j)) }
%Y A108956 Sequence in context: A052146 A097882 A108955 this_sequence A108037 A055679 A056172
%Y A108956 Adjacent sequences: A108953 A108954 A108955 this_sequence A108957 A108958 A108959
%K A108956 easy,nonn
%O A108956 2,2
%A A108956 Cino Hilliard (hillcino368(AT)hotmail.com), Jul 22 2005
%I A108037
%S A108037 0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,5,5,5,5,5,5,8,8,8,8,8,8,8,13,13,13,13,13,
%T A108037 13,13,13,21,21,21,21,21,21,21,21,21,34,34,34,34,34,34,34,34,34,34,55,55,
%U A108037 55,55,55,55,55,55,55,55,55,89,89,89,89,89,89,89,89,89,89,89,89,144,144
%N A108037 Triangle read by rows: n-th row is n-th nonzero Fibonacci number repeated n+1 times.
%F A108037 G.f.: x*(1+y-x*y)/((1-x-x^2)*(1-x*y-x^2*y^2)). [U coordinates]
%e A108037 0; 1,1; 1,1,1; 2,2,2,2; 3,3,3,3,3; 5,5,5,5,5,5; ...
%Y A108037 Cf. A039913, A108036, A108035.
%Y A108037 Sequence in context: A097882 A108955 A108956 this_sequence A055679 A056172 A091373
%Y A108037 Adjacent sequences: A108034 A108035 A108036 this_sequence A108038 A108039 A108040
%K A108037 nonn,tabl
%O A108037 0,7
%A A108037 njas, Jun 01 2005
%I A055679
%S A055679 0,0,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,7,7,
%T A055679 7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,11,11,11,11,
%U A055679 11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,13,13,13,13,14,14,14
%N A055679 Number of distinct prime factors of EulerPhi(n!).
%C A055679 Number of distinct prime factors of n! and Phi(n!) are respectively Pi(n) and Pi(Floor(n/2)).
%F A055679 A001221(A000010(A000142(n)))=A001221(A048855(n)))
%F A055679 a(n)=PrimePi[Floor[n/2]].
%Y A055679 Cf. A000142, A000010, A000720, A001221, A055718.
%Y A055679 Sequence in context: A108955 A108956 A108037 this_sequence A056172 A091373 A008621
%Y A055679 Adjacent sequences: A055676 A055677 A055678 this_sequence A055680 A055681 A055682
%K A055679 nonn
%O A055679 1,6
%A A055679 Labos E. (labos(AT)ana.sote.hu), Jul 11 2000
%I A056172
%S A056172 0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,7,7,
%T A056172 7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,11,11,11,11,
%U A056172 11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,13,13,13,13,14,14,14
%N A056172 Number of non-unitary prime divisors of n!.
%C A056172 A non-unitary prime divisor for n! is not larger than n/2. a(n)=PrimePi[n/2]
%F A056172 A prime divisor of x is not unitary iff its exponent is at least 2 in prime power factorization of x. In general GCD[p, x/p]=1 or p. Cases are counted when GCD[p, n/p]>1.
%e A056172 10!=2.2.2.2.2.2.2.2.3.3.3.3.5.5.7 The non-unitary prime divisors is 2,3,5 because their exponents exceed 1, so a(10)=3, while 10! has only 5 unitary prime divisor.
%Y A056172 A001221, A034444, A000720, A048105, A048656, A048657.
%Y A056172 Sequence in context: A108956 A108037 A055679 this_sequence A091373 A008621 A002265
%Y A056172 Adjacent sequences: A056169 A056170 A056171 this_sequence A056173 A056174 A056175
%K A056172 nonn
%O A056172 1,6
%A A056172 Labos E. (labos(AT)ana.sote.hu), Jul 27 2000
%I A091373
%S A091373 0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,6,7,7,7,7,7,7,7,8,8,
%T A091373 8,8,9,9,9,9,9,9,9,10,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,
%U A091373 13,14,14,14,14,14,14,14,14,14,14,14,15,16,16,16,16,16,16,16,17,17
%N A091373 Number of numbers <= n having exactly as many prime factors as the value of their smallest prime factor.
%C A091373 a(n) = #{m: A001222(m)=A020639(m), m<=n};
%C A091373 A091372(n) + a(n) + A091374(n) = n.
%Y A091373 Cf. A091376, A091371.
%Y A091373 Sequence in context: A108037 A055679 A056172 this_sequence A008621 A002265 A110655
%Y A091373 Adjacent sequences: A091370 A091371 A091372 this_sequence A091374 A091375 A091376
%K A091373 nonn
%O A091373 1,6
%A A091373 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jan 04 2004
%I A008621
%S A008621 1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,9,9,9,
%T A008621 9,10,10,10,10,11,11,11,11,12,12,12,12,13,13,13,13,14,14,14,14,15,15,15,
%U A008621 15,16,16,16,16,17,17,17,17,18,18,18,18,19,19
%N A008621 Expansion of 1/((1-x)*(1-x^4)).
%C A008621 Arises from Gleason's theorem on self-dual codes: 1/((1-x^2)*(1-x^8)) is the Molien series for the real 2-dimensional Clifford group (a dihedral group of order 16) of genus 1.
%D A008621 D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
%D A008621 F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes, 1977, Chapter 19, Problem 3, p. 602.
%H A008621 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 211
%H A008621 Index entries for Molien series
%F A008621 a(n)= floor((n+3)/4), n>0;
%t A008621 Table[Floor[(n + 3)/4], {n, 1, 80}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 03 2006
%Y A008621 Cf. A008718, A024186, A110160, A110868, A110869, A110876, A110880.
%Y A008621 Cf. A008620, A002265.
%Y A008621 a(n)=A010766(n+4,4).
%Y A008621 Sequence in context: A055679 A056172 A091373 this_sequence A002265 A110655 A075245
%Y A008621 Adjacent sequences: A008618 A008619 A008620 this_sequence A008622 A008623 A008624
%K A008621 nonn,easy,nice
%O A008621 0,5
%A A008621 njas
%E A008621 More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 03 2006
%I A002265
%S A002265 0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,
%T A002265 8,8,8,9,9,9,9,10,10,10,10,11,11,11,11,12,12,12,12,13,13,13,13,14,
%U A002265 14,14,14,15,15,15,15,16,16,16,16,17,17,17,17,18,18,18,18,19,19,19,19
%N A002265 Integers repeated 4 times.
%C A002265 For n>=1 and i=sqrt(-1) let F(n) the n X n matrix of the Discrete Fourier Transform (DFT) whose element (j,k) equals exp(-2*pi*i*(j-1)*(k-1)/n)/sqrt(n). The multiplicities of the four eigenvalues 1, i, -1, -i of F(n) are a(n+4), a(n-1), a(n+2), a(n+1), hence a(n+4) + a(n-1) + a(n+2) + a(n+1) = n for n>=1. E.g. the multiplicities of the eigenvalues 1, i, -1, -i of the DFT-matrix F(4) are a(8)=2, a(3)=0, a(6)=1, a(5)=1, summing up to 4. - Franz Vrabec (franz.vrabec(AT)planetuniqa.at), Jan 21 2005
%D A002265 V. Cizek, Discrete Fourier Transforms and their Applications, Adam Hilger, Bristol 1986, p. 61.
%D A002265 J. H. McClellan, T. W. Parks, Eigenvalue and Eigenvector Decomposition of the Discrete Fourier Transform, IEEE Trans. Audio and Electroacoust., Vol. AU-20, No.1, March 1972, pp. 66-74.
%F A002265 a(n) = floor(n/4), n>=0;
%Y A002265 Cf. A008621.
%Y A002265 Zero followed by partial sums of A011765.
%Y A002265 Sequence in context: A056172 A091373 A008621 this_sequence A110655 A075245 A008652
%Y A002265 Adjacent sequences: A002262 A002263 A002264 this_sequence A002266 A002267 A002268
%K A002265 nonn,easy
%O A002265 0,9
%A A002265 njas
%I A110655
%S A110655 0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,9,9,
%T A110655 9,9,10,10,10,10,11,11,11,11,12,12,12,12,13,13,13,13,14,14,14,14,15,15,
%U A110655 15,15,16,16,16,16,17,17,17,17,18,18,18,18,19,19,19,19,20,20,20,20,21
%N A110655 A110654(A110654(n)).
%C A110655 a(n) = A008621(n+1) = A002265(n+3);
%C A110655 A110656(n) = A110654(a(n)) = a(A110654(n));
%F A110655 a(n) = ceiling(n/4).
%Y A110655 Cf. A110657.
%Y A110655 Sequence in context: A091373 A008621 A002265 this_sequence A075245 A008652 A091226
%Y A110655 Adjacent sequences: A110652 A110653 A110654 this_sequence A110656 A110657 A110658
%K A110655 nonn
%O A110655 0,6
%A A110655 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 05 2005
%I A075245
%S A075245 1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,9,9,9,9,10,
%T A075245 10,10,10,11,11,11,11,12,12,12,12,13,14,13,13,14,14,14,14,15,15,15,15,
%U A075245 16,16,16,16,17,17,17,17,18,18,18,18,19,20,19,19,20,20,20,20,21,21,21
%N A075245 x-value of the solution (x,y,z) to 4/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z and having the largest z-value. The y and z components are in A075246 and A075247.
%C A075245 See A073101 for more details.
%t A075245 For[xLst={}; yLst={}; zLst={}; n=3, n<=100, n++, cnt=0; xr=n/4; If[IntegerQ[xr], x=xr+1, x=Ceiling[xr]]; While[yr=1/(4/n-1/x); If[IntegerQ[yr], y=yr+1, y=Ceiling[yr]]; cnt==0&&y>x, While[zr=1/(4/n-1/x-1/y); cnt==0&&zr>y, If[IntegerQ[zr], z=zr; cnt++; AppendTo[xLst, x]; AppendTo[yLst, y]; AppendTo[zLst, z]]; y++ ]; x++ ]]; xLst
%Y A075245 Cf. A073101, A075246, A075247.
%Y A075245 Sequence in context: A008621 A002265 A110655 this_sequence A008652 A091226 A097913
%Y A075245 Adjacent sequences: A075242 A075243 A075244 this_sequence A075246 A075247 A075248
%K A075245 hard,nice,nonn
%O A075245 3,2
%A A075245 T. D. Noe (noe(AT)sspectra.com), Sep 10 2002
%I A008652
%S A008652 1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8,8,8,8,10,10,
%T A008652 10,10,12,12,12,12,15,15,15,15,18,18,18,18,21,21,21,21,
%U A008652 24,24,24,24,28,28,28,28,32,32,32,32,36,36,36,36,40,40
%N A008652 Molien series for group of 3 X 3 upper triangular matrices over GF( 4 ).
%D A008652 D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 105.
%H A008652 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 220
%H A008652 Index entries for Molien series
%p A008652 1/(1-x)/(1-x^4)/(1-x^16)
%Y A008652 Sequence in context: A002265 A110655 A075245 this_sequence A091226 A097913 A029269
%Y A008652 Adjacent sequences: A008649 A008650 A008651 this_sequence A008653 A008654 A008655
%K A008652 nonn,easy,nice
%O A008652 0,5
%A A008652 njas
%I A091226
%S A091226 0,0,1,2,2,2,2,3,3,3,3,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,
%T A091226 8,8,9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,13,
%U A091226 13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,16,16,16
%N A091226 Number of irreducible GF(2)[X]-polynomials in range [0,n].
%C A091226 Analogous to A000720.
%H A091226 A. Karttunen, Scheme-program for computing this sequence.
%H A091226 Index entries for sequences operating on GF(2)[X]-polynomials
%Y A091226 Partial sums of A091225. A062692(n) = a(2^n).
%Y A091226 Sequence in context: A110655 A075245 A008652 this_sequence A097913 A029269 A088004
%Y A091226 Adjacent sequences: A091223 A091224 A091225 this_sequence A091227 A091228 A091229
%K A091226 nonn
%O A091226 0,4
%A A091226 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Jan 03 2004
%I A097913
%S A097913 1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,5,5,6,6,6,6,9,9,10,10,11,11,12,12,
%T A097913 15,15,16,16,19,19,20,20,23,23,26,26,29,29,30,30,36,36,39,39,42,42,45,45,
%U A097913 51,51,54,54,60,60,63,63,69,69,75,75,81,81,84,84,94,94,100,100,106,106
%N A097913 G.f.: (1+x^18)/((1-x)*(1-x^8)*(1-x^12)*(1-x^24)).
%C A097913 Conjectured Poincare series for genus 2 Siegel theta series of odd unimodular lattices.
%H A097913 G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
%Y A097913 Cf. A008718.
%Y A097913 Sequence in context: A075245 A008652 A091226 this_sequence A029269 A088004 A070548
%Y A097913 Adjacent sequences: A097910 A097911 A097912 this_sequence A097914 A097915 A097916
%K A097913 nonn
%O A097913 0,9
%A A097913 njas, Sep 04 2004
%I A029269
%S A029269 1,0,0,1,1,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,5,5,6,6,7,7,8,
%T A029269 8,9,9,11,11,12,13,14,14,16,16,17,18,20,20,22,23,25,25,
%U A029269 27,28,30,30,33,34,36,37,40,41,43,44,47,48,51,52,55,57
%N A029269 Expansion of 1/((1-x^3)(1-x^4)(1-x^10)(1-x^11)).
%Y A029269 Sequence in context: A008652 A091226 A097913 this_sequence A088004 A070548 A054893
%Y A029269 Adjacent sequences: A029266 A029267 A029268 this_sequence A029270 A029271 A029272
%K A029269 nonn
%O A029269 0,11
%A A029269 njas
%I A088004
%S A088004 1,1,1,1,1,2,2,2,2,3,3,3,3,4,5,5,5,5,5,5,6,7,7,7,7,8,8,8,8,7,7,7,8,9,10,
%T A088004 10,10,11,12,12,12,11,11,11,11,12,12,12,12,12,13,13,13,13,14,14,15,16,
%U A088004 16,16,16,17,17,17,18,17,17,17,18,17,17,17,17,18,18,18,19,18,18,18,18
%N A088004 Truncated "Mertens-function": values of -1 at primes are left out, i.e. summatory-Moebius when argument runs through nonprimes.
%F A088004 a(n)=A002321[n]-(-1).pi[n]=A002321(n)+A000720(n)
%e A088004 Since the principal source of negative excursions of Mertens-function is here eliminated, most probably this sequence increases ad infinitum albeit non-monotonically; decrease at squarefree numbers with odd number of p-divisors like 30,42,...
%t A088004 mer[x_] := mer[x-1]+MoebiusMu[x]; mer[0]=0; $RecursionLimit=1000; Table[mer[w]+PrimePi[w], {w, 1, 256}]
%Y A088004 Sequence in context: A091226 A097913
