cp's OEIS Frontend

To test if a number is a square, see Cohen, p. 40. - N. J. A. Sloane, Jun 19 2011

Zero followed by partial sums of A005408 (odd numbers). - Jeremy Gardiner, Aug 13 2002

Begin with n, add the next number, subtract the previous number and so on ending with subtracting a 1: a(n) = n + (n+1) - (n-1) + (n+2) - (n-2) + (n+3) - (n-3) + ... + (2n-1) - 1 = n^2. - Amarnath Murthy, Mar 24 2004

Sum of two consecutive triangular numbers A000217. - Lekraj Beedassy, May 14 2004

Numbers with an odd number of divisors: {d(n^2) = A048691(n); for the first occurrence of 2n + 1 divisors, see A071571(n)}. - Lekraj Beedassy, Jun 30 2004

See also A000037.

First sequence ever computed by electronic computer, on EDSAC, May 06 1949 (see Renwick link). - Russ Cox, Apr 20 2006

Numbers k such that the imaginary quadratic field Q(sqrt(-k)) has four units. - Marc LeBrun, Apr 12 2006

For n > 0: number of divisors of (n-1)th power of any squarefree semiprime: a(n) = A000005(A006881(k)^(n-1)); a(n) = A000005(A000400(n-1)) = A000005(A011557(n-1)) = A000005(A001023(n-1)) = A000005(A001024(n-1)). - Reinhard Zumkeller, Mar 04 2007

If a 2-set Y and an (n-2)-set Z are disjoint subsets of an n-set X then a(n-2) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 19 2007

Numbers a such that a^1/2 + b^1/2 = c^1/2 and a^2 + b = c. - Cino Hilliard, Feb 07 2008 (this comment needs clarification, Joerg Arndt, Sep 12 2013)

Numbers k such that the geometric mean of the divisors of k is an integer. - Ctibor O. Zizka, Jun 26 2008

Equals row sums of triangle A143470. Example: 36 = sum of row 6 terms: (23 + 7 + 3 + 1 + 1 + 1). - Gary W. Adamson, Aug 17 2008

Equals row sums of triangles A143595 and A056944. - Gary W. Adamson, Aug 26 2008

Number of divisors of 6^(n-1) for n > 0. - J. Lowell, Aug 30 2008

Denominators of Lyman spectrum of hydrogen atom. Numerators are A005563. A000290-A005563 = A000012. - Paul Curtz, Nov 06 2008

a(n) is the number of all partitions of the sum 2^2 + 2^2 + ... + 2^2, (n-1) times, into powers of 2. - Valentin Bakoev, Mar 03 2009

a(n) is the maximal number of squares that can be 'on' in an n X n board so that all the squares turn 'off' after applying the operation: in any 2 X 2 sub-board, a square turns from 'on' to 'off' if the other three are off. - Srikanth K S, Jun 25 2009

Zero together with the numbers k such that 2 is the number of perfect partitions of k. - Juri-Stepan Gerasimov, Sep 26 2009

Totally multiplicative sequence with a(p) = p^2 for prime p. - Jaroslav Krizek, Nov 01 2009

Satisfies A(x)/A(x^2), A(x) = A173277: (1, 4, 13, 32, 74, ...). - Gary W. Adamson, Feb 14 2010

Positive members are the integers with an odd number of odd divisors and an even number of even divisors. See also A120349, A120359, A181792, A181793, A181795. - Matthew Vandermast, Nov 14 2010

Besides the first term, this sequence is the denominator of Pi^2/6 = 1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + ... . - Mohammad K. Azarian, Nov 01 2011

Partial sums give A000330. - Omar E. Pol, Jan 12 2013

Drmota, Mauduit, and Rivat proved that the Thue-Morse sequence along the squares is normal; see A228039. - Jonathan Sondow, Sep 03 2013

a(n) can be decomposed into the sum of the four numbers [binomial(n, 1) + binomial(n, 2) + binomial(n-1, 1) + binomial(n-1, 2)] which form a "square" in Pascal's Triangle A007318, or the sum of the two numbers [binomial(n, 2) + binomial(n+1, 2)], or the difference of the two numbers [binomial(n+2, 3) - binomial(n, 3)]. - John Molokach, Sep 26 2013

In terms of triangular tiling, the number of equilateral triangles with side length 1 inside an equilateral triangle with side length n. - K. G. Stier, Oct 30 2013

Number of positive roots in the root systems of type B_n and C_n (when n > 1). - Tom Edgar, Nov 05 2013

Squares of squares (fourth powers) are also called biquadratic numbers: A000583. - M. F. Hasler, Dec 29 2013

For n > 0, a(n) is the largest integer k such that k^2 + n is a multiple of k + n. More generally, for m > 0 and n > 0, the largest integer k such that k^(2*m) + n is a multiple of k + n is given by k = n^(2*m). - Derek Orr, Sep 03 2014

For n > 0, a(n) is the number of compositions of n + 5 into n parts avoiding the part 2. - Milan Janjic, Jan 07 2016

a(n), for n >= 3, is also the number of all connected subtrees of a cycle graph, having n vertices. - Viktar Karatchenia, Mar 02 2016

On every sequence of natural continuous numbers with an even number of elements, the summatory of the second half of the sequence minus the summatory of the first half of the sequence is always a square. Example: Sequence from 61 to 70 has an even number of elements (10). Then 61 + 62 + 63 + 64 + 65 = 315; 66 + 67 + 68 + 69 + 70 = 340; 340 - 315 = 25. (n/2)^2 for n = number of elements. - César Aguilera, Jun 20 2016

On every sequence of natural continuous numbers from n^2 to (n+1)^2, the sum of the differences of pairs of elements of the two halves in every combination possible is always (n+1)^2. - César Aguilera, Jun 24 2016

Suppose two circles with radius 1 are tangent to each other as well as to a line not passing through the point of tangency. Create a third circle tangent to both circles as well as the line. If this process is continued, a(n) for n > 0 is the reciprocals of the radii of the circles, beginning with the largest circle. - Melvin Peralta, Aug 18 2016

Does not satisfy Benford's law [Ross, 2012]. - N. J. A. Sloane, Feb 08 2017

Numerators of the solution to the generalization of the Feynman triangle problem, with an offset of 2. If each vertex of a triangle is joined to the point (1/p) along the opposite side (measured say clockwise), then the area of the inner triangle formed by these lines is equal to (p - 2)^2/(p^2 - p + 1) times the area of the original triangle, p > 2. For example, when p = 3, the ratio of the areas is 1/7. The denominators of the ratio of the areas is given by A002061. [Cook & Wood, 2004] - Joe Marasco, Feb 20 2017

Equals row sums of triangle A004737, n >= 1. - Martin Michael Musatov, Nov 07 2017

Right-hand side of the binomial coefficient identity Sum_{k = 0..n} (-1)^(n+k+1)*binomial(n,k)*binomial(n + k,k)*(n - k) = n^2. - Peter Bala, Jan 12 2022

Conjecture: For n>0, min{k such that there exist subsets A,B of {0,1,2,...,a(n)-1} such that |A|=|B|=k and A+B contains {0,1,2,...,a(n)-1}} = n. - Michael Chu, Mar 09 2022

Number of 3-permutations of n elements avoiding the patterns 132, 213, 321. See Bonichon and Sun. - Michel Marcus, Aug 20 2022

Number of intercalates in cyclic Latin squares of order 2n (cyclic Latin squares of odd order do not have intercalates). - Eduard I. Vatutin, Feb 15 2024

a(n) is the number of ternary strings of length n with at most one 0, exactly one 1, and no restriction on the number of 2's. For example, a(3)=9, consisting of the 6 permutations of the string 102 and the 3 permutations of the string 122. - Enrique Navarrete, Mar 12 2025

1 together with the numbers that are products of distinct primes.

Also smallest sequence with the property that a(m)*a(k) is never a square for k != m. - Ulrich Schimke (ulrschimke(AT)aol.com), Dec 12 2001

Numbers k such that there is only one Abelian group with k elements, the cyclic group of order k (the numbers such that A000688(k) = 1). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 25 2001

Numbers k such that A007913(k) > phi(k). - Benoit Cloitre, Apr 10 2002

a(n) is the smallest m with exactly n squarefree numbers <= m. - Amarnath Murthy, May 21 2002

k is squarefree <=> k divides prime(k)# where prime(k)# = product of first k prime numbers. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 30 2004

Numbers k such that omega(k) = Omega(k) = A072047(k). - Lekraj Beedassy, Jul 11 2006

The LCM of any finite subset is in this sequence. - Lekraj Beedassy, Jul 11 2006

This sequence and the Beatty Pi^2/6 sequence (A059535) are "incestuous": the first 20000 terms are bounded within (-9, 14). - Ed Pegg Jr, Jul 22 2008

Let us introduce a function D(n) = sigma_0(n)/2^(alpha(1) + ... + alpha(r)), sigma_0(n) number of divisors of n (A000005), prime factorization of n = p(1)^alpha(1) * ... * p(r)^alpha(r), alpha(1) + ... + alpha(r) is sequence (A001222). Function D(n) splits the set of positive integers into subsets, according to the value of D(n). Squarefree numbers (A005117) has D(n)=1, other numbers are "deviated" from the squarefree ideal and have 0 < D(n) < 1. For D(n)=1/2 we have A048109, for D(n)=3/4 we have A060687. - Ctibor O. Zizka, Sep 21 2008

Numbers k such that gcd(k,k')=1 where k' is the arithmetic derivative (A003415) of k. - Giorgio Balzarotti, Apr 23 2011

Numbers k such that A007913(k) = core(k) = k. - Franz Vrabec, Aug 27 2011

Numbers k such that sqrt(k) cannot be simplified. - Sean Loughran, Sep 04 2011

Indices m where A057918(m)=0, i.e., positive integers m for which there are no integers k in {1,2,...,m-1} such that k*m is a square. - John W. Layman, Sep 08 2011

It appears that these are numbers j such that Product_{k=1..j} (prime(k) mod j) = 0 (see Maple code). - Gary Detlefs, Dec 07 2011. - This is the same claim as Mohammed Bouayoun's Mar 30 2004 comment above. To see why it holds: Primorial numbers, A002110, a subsequence of this sequence, are never divisible by any nonsquarefree number, A013929, and on the other hand, the index of the greatest prime dividing any n is less than n. Cf. A243291. - Antti Karttunen, Jun 03 2014

Conjecture: For each n=2,3,... there are infinitely many integers b > a(n) such that Sum_{k=1..n} a(k)*b^(k-1) is prime, and the smallest such an integer b does not exceed (n+3)*(n+4). - Zhi-Wei Sun, Mar 26 2013

The probability that a random natural number belongs to the sequence is 6/Pi^2, A059956 (see Cesàro reference). - Giorgio Balzarotti, Nov 21 2013

Booker, Hiary, & Keating give a subexponential algorithm for testing membership in this sequence without factoring. - Charles R Greathouse IV, Jan 29 2014

Because in the factorizations into prime numbers these a(n) (n >= 2) have exponents which are either 0 or 1 one could call the a(n) 'numbers with a fermionic prime number decomposition'. The levels are the prime numbers prime(j), j >= 1, and the occupation numbers (exponents) e(j) are 0 or 1 (like in Pauli's exclusion principle). A 'fermionic state' is then denoted by a sequence with entries 0 or 1, where, except for the zero sequence, trailing zeros are omitted. The zero sequence stands for a(1) = 1. For example a(5) = 6 = 2^1*3^1 is denoted by the 'fermionic state' [1, 1], a(7) = 10 by [1, 0, 1]. Compare with 'fermionic partitions' counted in A000009. - Wolfdieter Lang, May 14 2014

From Vladimir Shevelev, Nov 20 2014: (Start)

The following is an Eratosthenes-type sieve for squarefree numbers. For integers > 1:

1) Remove even numbers, except for 2; the minimal non-removed number is 3.

2) Replace multiples of 3 removed in step 1, and remove multiples of 3 except for 3 itself; the minimal non-removed number is 5.

3) Replace multiples of 5 removed as a result of steps 1 and 2, and remove multiples of 5 except for 5 itself; the minimal non-removed number is 6.

4) Replace multiples of 6 removed as a result of steps 1, 2 and 3 and remove multiples of 6 except for 6 itself; the minimal non-removed number is 7.

5) Repeat using the last minimal non-removed number to sieve from the recovered multiples of previous steps.

Proof. We use induction. Suppose that as a result of the algorithm, we have found all squarefree numbers less than n and no other numbers. If n is squarefree, then the number of its proper divisors d > 1 is even (it is 2^k - 2, where k is the number of its prime divisors), and, by the algorithm, it remains in the sequence. Otherwise, n is removed, since the number of its squarefree divisors > 1 is odd (it is 2^k-1).

(End)

The lexicographically least sequence of integers > 1 such that each entry has an even number of proper divisors occurring in the sequence (that's the sieve restated). - Glen Whitney, Aug 30 2015

0 is nonsquarefree because it is divisible by any square. - Jon Perry, Nov 22 2014, edited by M. F. Hasler, Aug 13 2015

The Heinz numbers of partitions with distinct parts. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} prime(j) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] the Heinz number is 2*2*3*7*29 = 2436. The number 30 (= 2*3*5) is in the sequence because it is the Heinz number of the partition [1,2,3]. - Emeric Deutsch, May 21 2015

It is possible for 2 consecutive terms to be even; for example a(258)=422 and a(259)=426. - Thomas Ordowski, Jul 21 2015. [These form a subsequence of A077395 since their product is divisible by 4. - M. F. Hasler, Aug 13 2015]

There are never more than 3 consecutive terms. Runs of 3 terms start at 1, 5, 13, 21, 29, 33, ... (A007675). - Ivan Neretin, Nov 07 2015

a(n) = product of row n in A265668. - Reinhard Zumkeller, Dec 13 2015

Numbers without excess, i.e., numbers k such that A001221(k) = A001222(k). - Juri-Stepan Gerasimov, Sep 05 2016

Numbers k such that b^(phi(k)+1) == b (mod k) for every integer b. - Thomas Ordowski, Oct 09 2016

Boreico shows that the set of square roots of the terms of this sequence is linearly independent over the rationals. - Jason Kimberley, Nov 25 2016 (reference found by Michael Coons).

Numbers k such that A008836(k) = A008683(k). - Enrique Pérez Herrero, Apr 04 2018

The prime zeta function P(s) "has singular points along the real axis for s=1/k where k runs through all positive integers without a square factor". See Wolfram link. - Maleval Francis, Jun 23 2018

Numbers k such that A007947(k) = k. - Kyle Wyonch, Jan 15 2021

The Schnirelmann density of the squarefree numbers is 53/88 (Rogers, 1964). - Amiram Eldar, Mar 12 2021

Comment from Isaac Saffold, Dec 21 2021: (Start)

Numbers k such that all groups of order k have a trivial Frattini subgroup [Dummit and Foote].

Let the group G have order n. If n is squarefree and n > 1, then G is solvable, and thus by Hall's Theorem contains a subgroup H_p of index p for all p | n. Each H_p is maximal in G by order considerations, and the intersection of all the H_p's is trivial. Thus G's Frattini subgroup Phi(G), being the intersection of G's maximal subgroups, must be trivial. If n is not squarefree, the cyclic group of order n has a nontrivial Frattini subgroup. (End)

Numbers for which the squarefree divisors (A206778) and the unitary divisors (A077610) are the same; moreover they are also the set of divisors (A027750). - Bernard Schott, Nov 04 2022

0 = A008683(a(n)) - A008836(a(n)) = A001615(a(n)) - A000203(a(n)). - Torlach Rush, Feb 08 2023

From Robert D. Rosales, May 20 2024: (Start)

Numbers n such that mu(n) != 0, where mu(n) is the Möbius function (A008683).

Solutions to the equation Sum_{d|n} mu(d)*sigma(d) = mu(n)*n, where sigma(n) is the sum of divisors function (A000203). (End)

a(n) is the smallest root of x = 1 + Sum_{k=1..n-1} floor(sqrt(x/a(k))) greater than a(n-1). - Yifan Xie, Jul 10 2024

Number k such that A001414(k) = A008472(k). - Torlach Rush, Apr 14 2025

To elaborate on the formula from Greathouse (2018), the maximum of a(n) - floor(n*Pi^2/6 + sqrt(n)/17) is 10 at indices n = 48715, 48716, 48721, and 48760. The maximum is 11, at the same indices, if floor is taken individually for the two addends and the square root. If the value is rounded instead, the maximum is 9 at 10 indices between 48714 and 48765. - M. F. Hasler, Aug 08 2025

Sometimes called Archimedes's constant.

Ratio of a circle's circumference to its diameter.

Also area of a circle with radius 1.

Also surface area of a sphere with diameter 1.

A useful mnemonic for remembering the first few terms: How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics ...

Also ratio of surface area of sphere to one of the faces of the circumscribed cube. Also ratio of volume of a sphere to one of the six inscribed pyramids in the circumscribed cube. - Omar E. Pol, Aug 09 2012

Also surface area of a quarter of a sphere of radius 1. - Omar E. Pol, Oct 03 2013

Also the area under the peak-shaped even function f(x)=1/cosh(x). Proof: for the upper half of the integral, write f(x) = (2*exp(-x))/(1+exp(-2x)) = 2*Sum_{k>=0} (-1)^k*exp(-(2k+1)*x) and integrate term by term from zero to infinity. The result is twice the Gregory series for Pi/4. - Stanislav Sykora, Oct 31 2013

A curiosity: a 144 X 144 magic square of 7th powers was recently constructed by Toshihiro Shirakawa. The magic sum = 3141592653589793238462643383279502884197169399375105, which is the concatenation of the first 52 digits of Pi. See the MultiMagic Squares link for details. - Christian Boyer, Dec 13 2013 [Comment revised by N. J. A. Sloane, Aug 27 2014]

x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 25 2013

Also diameter of a sphere whose surface area equals the volume of the circumscribed cube. - Omar E. Pol, Jan 13 2014

From Daniel Forgues, Mar 20 2015: (Start)

An interesting anecdote about the base-10 representation of Pi, with 3 (integer part) as first (index 1) digit:

358 0

359 3

360 6

361 0

362 0

And the circle is customarily subdivided into 360 degrees (although Pi radians yields half the circle)...

(End)

Sometimes referred to as Archimedes's constant, because the Greek mathematician computed lower and upper bounds of Pi by drawing regular polygons inside and outside a circle. In Germany it was called the Ludolphian number until the early 20th century after the Dutch mathematician Ludolph van Ceulen (1540-1610), who calculated up to 35 digits of Pi in the late 16th century. - Martin Renner, Sep 07 2016

As of the beginning of 2019 more than 22 trillion decimal digits of Pi are known. See the Wikipedia article "Chronology of computation of Pi". - Harvey P. Dale, Jan 23 2019

On March 14, 2019, Emma Haruka Iwao announced the calculation of 31.4 trillion digits of Pi using Google Cloud's infrastructure. - David Radcliffe, Apr 10 2019

Also volume of three quarters of a sphere of radius 1. - Omar E. Pol, Aug 16 2019

On August 5, 2021, researchers from the University of Applied Sciences of the Grisons in Switzerland announced they had calculated 62.8 trillion digits. Guinness World Records has not verified this yet. - Alonso del Arte, Aug 23 2021

The Hermite-Lindemann (1882) theorem states, that if z is a nonzero algebraic number, then e^z is a transcendent number. The transcendence of Pi then results from Euler's relation: e^(i*Pi) = -1. - Peter Luschny, Jul 21 2023

The 10000 words of the book "Not A Wake" by Michael Keith, written in Pilish, match in length the first 10000 digits of Pi. - Paolo Xausa, Aug 07 2025

Sometimes called Apéry's constant.

"A natural question is whether Zeta(3) is a rational multiple of Pi^3. This is not known, though in 1978 R. Apéry succeeded in proving that Zeta(3) is irrational. In Chapter 8 we pointed out that the probability that two random integers are relatively prime is 6/Pi^2, which is 1/Zeta(2). This generalizes to: The probability that k random integers are relatively prime is 1/Zeta(k) ... ." [Stan Wagon]

In 2001 Tanguy Rivoal showed that there are infinitely many odd (positive) integers at which zeta is irrational, including at least one value j in the range 5 <= j <= 21 (refined the same year by Zudilin to 5 <= j <= 11), at which zeta(j) is irrational. See the Rivoal link for further information and references.

The reciprocal of this constant is the probability that three integers chosen randomly using uniform distribution are relatively prime. - Joseph Biberstine (jrbibers(AT)indiana.edu), Apr 13 2005

Also the value of zeta(1,2), the double zeta-function of arguments 1 and 2. - R. J. Mathar, Oct 10 2011

Also the length of minimal spanning tree for large complete graph with uniform random edge lengths between 0 and 1, cf. link to John Baez's comment. - M. F. Hasler, Sep 26 2017

Sum of the inverses of the cubes (A000578). - Michael B. Porter, Nov 27 2017

This number is the average value of sigma_2(n)/n^2 where sigma_2(n) is the sum of the squares of the divisors of n. - Dimitri Papadopoulos, Jan 07 2022

b(n) = a(n+2) is the number of multigraphs with loops on 2 nodes with n edges [so g.f. for b(n) is 1/((1-x)^2*(1-x^2))]. Also number of 2-covers of an n-set; also number of 2 X n binary matrices with no zero columns up to row and column permutation. - Vladeta Jovovic, Jun 08 2000

a(n) is also the maximal number of edges that a triangle-free graph of n vertices can have. For n = 2m, the maximum is achieved by the bipartite graph K(m, m); for n = 2m + 1, the maximum is achieved by the bipartite graph K(m, m + 1). - Avi Peretz (njk(AT)netvision.net.il), Mar 18 2001

a(n) is the number of arithmetic progressions of 3 terms and any mean which can be extracted from the set of the first n natural numbers (starting from 1). - Santi Spadaro, Jul 13 2001

This is also the order dimension of the (strong) Bruhat order on the Coxeter group A_{n-1} (the symmetric group S_n). - Nathan Reading (reading(AT)math.umn.edu), Mar 07 2002

Let M_n denote the n X n matrix m(i,j) = 2 if i = j; m(i, j) = 1 if (i+j) is even; m(i, j) = 0 if i + j is odd, then a(n+2) = det M_n. - Benoit Cloitre, Jun 19 2002

Sums of pairs of neighboring terms are triangular numbers in increasing order. - Amarnath Murthy, Aug 19 2002

Also, from the starting position in standard chess, minimum number of captures by pawns of the same color to place n of them on the same file (column). Beyond a(6), the board and number of pieces available for capture are assumed to be extended enough to accomplish this task. - Rick L. Shepherd, Sep 17 2002

For example, a(2) = 1 and one capture can produce "doubled pawns", a(3) = 2 and two captures is sufficient to produce tripled pawns, etc. (Of course other, uncounted, non-capturing pawn moves are also necessary from the starting position in order to put three or more pawns on a given file.) - Rick L. Shepherd, Sep 17 2002

Terms are the geometric mean and arithmetic mean of their neighbors alternately. - Amarnath Murthy, Oct 17 2002

Maximum product of two integers whose sum is n. - Matthew Vandermast, Mar 04 2003

a(n+2) gives number of non-symmetric partitions of n into at most 3 parts, with zeros used as padding. E.g., a(7) = 12 because we can write 5 = 5 + 0 + 0 = 0 + 5 + 0 = 4 + 1 + 0 = 1 + 4 + 0 = 1 + 0 + 4 = 3 + 2 + 0 = 2 + 3 + 0 = 2 + 0 + 3 = 2 + 2 + 1 = 2 + 1 + 2 = 3 + 1 + 1 = 1 + 3 + 1. - Jon Perry, Jul 08 2003

a(n-1) gives number of distinct elements greater than 1 of non-symmetric partitions of n into at most 3 parts, with zeros used as padding, appear in the middle. E.g., 5 = 5 + 0 + 0 = 0 + 5 + 0 = 4 + 1 + 0 = 1 + 4 + 0 = 1 + 0 + 4 = 3 + 2 + 0 = 2 + 3 + 0 = 2 + 0 + 3 = 2 + 2 + 1 = 2 + 1 + 2 = 3 + 1 + 1 = 1 + 3 + 1. Of these, 050, 140, 320, 230, 221, 131 qualify and a(4) = 6. - Jon Perry, Jul 08 2003

Union of square numbers (A000290) and oblong numbers (A002378). - Lekraj Beedassy, Oct 02 2003

Conjectured size of the smallest critical set in a Latin square of order n (true for n <= 8). - Richard Bean, Jun 12 2003 and Nov 18 2003

a(n) gives number of maximal strokes on complete graph K_n, when edges on K_n can be assigned directions in any way. A "stroke" is a locally maximal directed path on a directed graph. Examples: n = 3, two strokes can exist, "x -> y -> z" and " x -> z", so a(3) = 2. n = 4, four maximal strokes exist, "u -> x -> z" and "u -> y" and "u -> z" and "x -> y -> z", so a(4) = 4. - Yasutoshi Kohmoto, Dec 20 2003

Number of symmetric Dyck paths of semilength n+1 and having three peaks. E.g., a(4) = 4 because we have U*DUUU*DDDU*D, UU*DUU*DDU*DD, UU*DDU*DUU*DD and UUU*DU*DU*DDD, where U = (1, 1), D = (1, -1) and * indicates a peak. - Emeric Deutsch, Jan 12 2004

Number of valid inequalities of the form j + k < n + 1, where j and k are positive integers, j <= k, n >= 0. - Rick L. Shepherd, Feb 27 2004

See A092186 for another application.

Also, the number of nonisomorphic transversal combinatorial geometries of rank 2. - Alexandr S. Radionov (rasmailru(AT)mail.ru), Jun 02 2004

a(n+1) is the transform of n under the Riordan array (1/(1-x^2), x). - Paul Barry, Apr 16 2005

1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ... specifies the largest number of copies of any of the gifts you receive on the n-th day in the "Twelve Days of Christmas" song. For example, on the fifth day of Christmas, you have 9 French hens. - Alonso del Arte, Jun 17 2005

a(n+1) is the number of noncongruent integer-sided triangles with largest side n. - David W. Wilson [Comment corrected Sep 26 2006]

A quarter-square table can be used to multiply integers since n*m = a(n+m) - a(n-m) for all integer n, m. - Michael Somos, Oct 29 2006

The sequence is the size of the smallest strong critical set in a Latin square of order n. - G.H.J. van Rees (vanrees(AT)cs.umanitoba.ca), Feb 16 2007

Maximal number of squares (maximal area) in a polyomino with perimeter 2n. - Tanya Khovanova, Jul 04 2007

For n >= 3 a(n-1) is the number of bracelets with n+3 beads, 2 of which are red, 1 of which is blue. - Washington Bomfim, Jul 26 2008

Equals row sums of triangle A122196. - Gary W. Adamson, Nov 29 2008

Also a(n) is the number of different patterns of a 2-colored 3-partition of n. - Ctibor O. Zizka, Nov 19 2014

Also a(n-1) = C(((n+(n mod 2))/2), 2) + C(((n-(n mod 2))/2), 2), so this is the second diagonal of A061857 and A061866, and each even-indexed term is the average of its two neighbors. - Antti Karttunen

Equals triangle A171608 * ( 1, 2, 3, ...). - Gary W. Adamson, Dec 12 2009

a(n) gives the number of nonisomorphic faithful representations of the Symmetric group S_3 of dimension n. Any faithful representation of S_3 must contain at least one copy of the 2-dimensional irrep, along with any combination of the two 1-dimensional irreps. - Andrew Rupinski, Jan 20 2011

a(n+2) gives the number of ways to make change for "c" cents, letting n = floor(c/5) to account for the 5-repetitive nature of the task, using only pennies, nickels and dimes (see A187243). - Adam Sasson, Mar 07 2011

a(n) belongs to the sequence if and only if a(n) = floor(sqrt(a(n))) * ceiling(sqrt(a(n))), that is, a(n) = k^2 or a(n) = k*(k+1), k >= 0. - Daniel Forgues, Apr 17 2011

a(n) is the sum of the positive integers < n that have the opposite parity as n.

Deleting the first 0 from the sequence results in a sequence b = 0, 1, 2, 4, ... such that b(n) is sum of the positive integers <= n that have the same parity as n. The sequence b(n) is the additive counterpart of the double factorial. - Peter Luschny, Jul 06 2011

Third outer diagonal of Losanitsch's Triangle, A034851. - Fred Daniel Kline, Sep 10 2011

Written as a(1) = 1, a(n) = a(n-1) + ceiling (a(n-1)) this is to ceiling as A002984 is to floor, and as A033638 is to round. - Jonathan Vos Post, Oct 08 2011

a(n-2) gives the number of distinct graphs with n vertices and n regions. - Erik Hasse, Oct 18 2011

Construct the n-th row of Pascal's triangle (A007318) from the preceding row, starting with row 0 = 1. a(n) counts the total number of additions required to compute the triangle in this way up to row n, with the restrictions that copying a term does not count as an addition, and that all additions not required by the symmetry of Pascal's triangle are replaced by copying terms. - Douglas Latimer, Mar 05 2012

a(n) is the sum of the positive differences of the parts in the partitions of n+1 into exactly 2 parts. - Wesley Ivan Hurt, Jan 27 2013

a(n) is the maximum number of covering relations possible in an n-element graded poset. For n = 2m, this bound is achieved for the poset with two sets of m elements, with each point in the "upper" set covering each point in the "lower" set. For n = 2m+1, this bound is achieved by the poset with m nodes in an upper set covering each of m+1 nodes in a lower set. - Ben Branman, Mar 26 2013

a(n+2) is the number of (integer) partitions of n into 2 sorts of 1's and 1 sort of 2's. - Joerg Arndt, May 17 2013

Alternative statement of Oppermann's conjecture: For n>2, there is at least one prime between a(n) and a(n+1). - Ivan N. Ianakiev, May 23 2013. [This conjecture was mentioned in A220492, A222030. - Omar E. Pol, Oct 25 2013]

For any given prime number, p, there are an infinite number of a(n) divisible by p, with those a(n) occurring in evenly spaced clusters of three as a(n), a(n+1), a(n+2) for a given p. The divisibility of all a(n) by p and the result are given by the following equations, where m >= 1 is the cluster number for that p: a(2m*p)/p = p*m^2 - m; a(2m*p + 1)/p = p*m^2; a(2m*p + 2)/p = p*m^2 + m. The number of a(n) instances between clusters is 2*p - 3. - Richard R. Forberg, Jun 09 2013

Apart from the initial term this is the elliptic troublemaker sequence R_n(1,2) in the notation of Stange (see Table 1, p.16). For other elliptic troublemaker sequences R_n(a,b) see the cross references below. - Peter Bala, Aug 08 2013

a(n) is also the total number of twin hearts patterns (6c4c) packing into (n+1) X (n+1) coins, the coins left is A042948 and the voids left is A000982. See illustration in links. - Kival Ngaokrajang, Oct 24 2013

Partitions of 2n into parts of size 1, 2 or 4 where the largest part is 4, i.e., A073463(n,2). - Henry Bottomley, Oct 28 2013

a(n+1) is the minimum length of a sequence (of not necessarily distinct terms) that guarantees the existence of a (not necessarily consecutive) subsequence of length n in which like terms appear consecutively. This is also the minimum cardinality of an ordered set S that ensures that, given any partition of S, there will be a subset T of S so that the induced subpartition on T avoids the pattern ac/b, where a < b < c. - Eric Gottlieb, Mar 05 2014

Also the number of elements of the list 1..n+1 such that for any two elements {x,y} the integer (x+y)/2 lies in the range ]x,y[. - Robert G. Wilson v, May 22 2014

Number of lattice points (x,y) inside the region of the coordinate plane bounded by x <= n, 0 < y <= x/2. For a(11)=30 there are exactly 30 lattice points in the region below:

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0 0 1 2 4 6 9 12 16 20 25 30 .. a(n) - Wesley Ivan Hurt, Oct 26 2014

a(n+1) is the greatest integer k for which there exists an n x n matrix M of nonnegative integers with every row and column summing to k, such that there do not exist n entries of M, all greater than 1, and no two of these entries in the same row or column. - Richard Stanley, Nov 19 2014

In a tiling of the triangular shape T_N with row length k for row k = 1, 2, ..., N >= 1 (or, alternatively row length N = 1-k for row k) with rectangular tiles, there can appear rectangles (i, j), N >= i >= j >= 1, of a(N+1) types (and their transposed shapes obtained by interchanging i and j). See the Feb 27 2004 comment above from Rick L. Shepherd. The motivation to look into this came from a proposal of Kival Ngaokrajang in A247139. - Wolfdieter Lang, Dec 09 2014

Every positive integer is a sum of at most four distinct quarter-squares; see A257018. - Clark Kimberling, Apr 15 2015

a(n+1) gives the maximal number of distinct elements of an n X n matrix which is symmetric (w.r.t. the main diagonal) and symmetric w.r.t. the main antidiagonal. Such matrices are called bisymmetric. See the Wikipedia link. - Wolfdieter Lang, Jul 07 2015

For 2^a(n+1), n >= 1, the number of binary bisymmetric n X n matrices, see A060656(n+1) and the comment and link by Dennis P. Walsh. - Wolfdieter Lang, Aug 16 2015

a(n) is the number of partitions of 2n+1 of length three with exactly two even entries (see below example). - John M. Campbell, Jan 29 2016

a(n) is the sum of the asymmetry degrees of all 01-avoiding binary words of length n. The asymmetry degree of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. a(6) = 9 because the 01-avoiding binary words of length 6 are 000000, 100000, 110000, 111000, 111100, 111110, and 111111, and the sum of their asymmetry degrees is 0 + 1 + 2 + 3 + 2 + 1 + 0 = 9. Equivalently, a(n) = Sum_{k>=0} k*A275437(n,k). - Emeric Deutsch, Aug 15 2016

a(n) is the number of ways to represent all the integers in the interval [3,n+1] as the sum of two distinct natural numbers. E.g., a(7)=12 as there are 12 different ways to represent all the numbers in the interval [3,8] as the sum of two distinct parts: 1+2=3, 1+3=4, 1+4=5, 1+5=6, 1+6=7, 1+7=8, 2+3=5, 2+4=6, 2+5=7, 2+6=8, 3+4=7, 3+5=8. - Anton Zakharov, Aug 24 2016

a(n+2) is the number of conjugacy classes of involutions (considering the identity as an involution) in the hyperoctahedral group C_2 wreath S_n. - Mark Wildon, Apr 22 2017

a(n+2) is the maximum number of pieces of a pizza that can be made with n cuts that are parallel or perpendicular to each other. - Anton Zakharov, May 11 2017

Also the matching number of the n X n black bishop graph. - Eric W. Weisstein, Jun 26 2017

The answer to a question posed by W. Mantel: a(n) is the maximum number of edges in an n-vertex triangle-free graph. Also solved by H. Gouwentak, J. Teixeira de Mattes, F. Schuh and W. A. Wythoff. - Charles R Greathouse IV, Feb 01 2018

Number of nonisomorphic outer planar graphs of order n >= 3, size n+2, and maximum degree 4. - Christian Barrientos and Sarah Minion, Feb 27 2018

Maximum area of a rectangle with perimeter 2n and sides of integer length. - André Engels, Jul 29 2018

Also the crossing number of the complete bipartite graph K_{3,n+1}. - Eric W. Weisstein, Sep 11 2018

a(n+2) is the number of distinct genotype frequency vectors possible for a sample of n diploid individuals at a biallelic genetic locus with a specified major allele. Such vectors are the lists of nonnegative genotype frequencies (n_AA, n_AB, n_BB) with n_AA + n_AB + n_BB = n and n_AA >= n_BB. - Noah A Rosenberg, Feb 05 2019

a(n+2) is the number of distinct real spectra (eigenvalues repeated according to their multiplicity) for an orthogonal n X n matrix. The case of an empty spectrum list is logically counted as one of those possibilities, when it exists. Thus a(n+2) is the number of distinct reduced forms (on the real field, in orthonormal basis) for elements in O(n). - Christian Devanz, Feb 13 2019

a(n) is the number of non-isomorphic asymmetric graphs that can be created by adding a single edge to a path on n+4 vertices. - Emma Farnsworth, Natalie Gomez, Herlandt Lino, and Darren Narayan, Jul 03 2019

a(n+1) is the number of integer triangles with largest side n. - James East, Oct 30 2019

a(n) is the number of nonempty subsets of {1,2,...,n} that contain exactly one odd and one even number. For example, for n=7, a(7)=12 and the 12 subsets are {1,2}, {1,4}, {1,6}, {2,3}, {2,5}, {2,7}, {3,4}, {3,6}, {4,5}, {4,7}, {5,6}, {6,7}. - Enrique Navarrete, Dec 16 2019

a(n+1) is also the n-th term of the Saind sequence (w_n){n>=1}, i.e., the infinite sequence caused by the entries of the queue of the degree sequences associated with the Saind arrays, as n increases. - _Giulia Palma, Jun 24 2020

Aside from the first two terms, a(n) enumerates the number of distinct normal ordered terms in the expansion of the differential operator (x + d/dx)^m associated to the Hermite polynomials and the Heisenberg-Weyl algebra. It also enumerates the number of distinct monomials in the bivariate polynomials corresponding to the partial sums of the series for cos(x+y) and sin(x+y). Cf. A344678. - Tom Copeland, May 27 2021

a(n) is the maximal number of negative products a_i * a_j (1 <= i <= j <= n), where all a_i are real numbers. - Logan Pipes, Jul 08 2021

From Allan Bickle, Dec 20 2021: (Start)

a(n) is the maximum product of the chromatic numbers of a graph of order n-1 and its complement. The extremal graphs are characterized in the papers of Finck (1968) and Bickle (2023).

a(n) is the maximum product of the degeneracies of a graph of order n+1 and its complement. The extremal graphs are characterized in the paper of Bickle (2012). (End)

a(n) is the maximum number m such that m white rooks and m black rooks can coexist on an n-1 X n-1 chessboard without attacking each other. - Aaron Khan, Jul 13 2022

Partial sums of A004526. - Bernard Schott, Jan 06 2023

a(n) is the number of 231-avoiding odd Grassmannian permutations of size n. - Juan B. Gil, Mar 10 2023

a(n) is the number of integer tuples (x,y) satisfying n + x + y >= 0, 25*n + x - 11*y >=0, 25*n - 11*x + y >=0, n + x + y == 0 (mod 12) , 25*n + x - 11*y == 0 (mod 5), 25*n - 11*x + y == 0 (mod 5) . For n=2, the sole solution is (x,y) = (0,0) and so a(2) = 1. For n = 3, the a(3) = 2 solutions are (-3, 2) and (2, -3). - Jeffery Opoku, Feb 16 2024

Let us consider triangles whose vertices are the centers of three squares constructed on the sides of a right triangle. a(n) is the integer part of the area of these triangles, taken without repetitions and in ascending order. See the illustration in the links. - Nicolay Avilov, Aug 05 2024

For n>=2, a(n) is the indendence number of the 2-token graph F_2(P_n) of the path graph P_n on n vertices. (Alternatively, as noted by Peter Munn, F_2(P_n) is the nXn square lattice, or grid, graph diminished by a cut across the diagonal.) - Miquel A. Fiol, Oct 05 2024

For n >= 1, also the lower matching number of the n-triangular honeycomb rook graph. - Eric W. Weisstein, Dec 14 2024

a(n-1) is also the minimal number of edges that a graph of n vertices must have such that any 3 vertices share at least one edge. - Ruediger Jehn, May 20 2025

a(n) is the number of edges of the antiregular graph A_n. This is the unique connected graph with n vertices and degrees 1 to n-1 (floor(n/2) repeated). - Allan Bickle, Jun 15 2025

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Numerators of coefficients in expansion of Sum_{n >= 1} x^n / (n*(1-x^n)) = Sum_{n >= 1} log(1/(1-x^n)).

The primes in this sequence, in order of appearance (without multiplicity), begin: 3, 7, 2, 13, 31, 5, 127. The first occurrence of prime(k) = a(n) for k = 1, 2, 3, ... is at n = 6, 2, 24, 4, 35640, 9, 297600, 588, ... - Jonathan Vos Post, Apr 02 2011

With amicable numbers, we have a(A002025(n)) = a(A002046(n)). - Michel Marcus, Dec 29 2013

Numerator of sigma(n)/n = A000203(n)/n. See A239578(n) - the smallest number k such that a(k) = n. - Jaroslav Krizek, Sep 23 2014

Also the rows of both triangles A237270 and A237593 interleaved.

Also, irregular triangle read by rows in which T(n,k) is the area of the k-th region (from left to right in ascending diagonal) of the n-th symmetric set of regions (from the top to the bottom in descending diagonal) in the two-dimensional diagram of the perspective view of the infinite stepped pyramid described in A245092 (see the diagram in the Links section).

The diagram of the symmetric representation of sigma is also the top view of the pyramid, see Links section. For more information about the diagram see also A237593 and A237270.

The number of cubes at the n-th level is also A024916(n), the sum of all divisors of all positive integers <= n.

Note that this pyramid is also a quarter of the pyramid described in A244050. Both pyramids have infinitely many levels.

Odd-indexed rows are also the rows of the irregular triangle A237270.

Even-indexed rows are also the rows of the triangle A237593.

Lengths of the odd-indexed rows are in A237271.

Lengths of the even-indexed rows give 2*A003056.

Row sums of the odd-indexed rows gives A000203, the sum of divisors function.

Row sums of the even-indexed rows give the positive even numbers (see A005843).

Row sums give A245092.

From the front view of the stepped pyramid emerges a geometric pattern which is related to A001227, the number of odd divisors of the positive integers.

The connection with the odd divisors of the positive integers is as follows: A261697 --> A261699 --> A237048 --> A235791 --> A237591 --> A237593 --> A237270 --> this sequence.

In a widely distributed May 2011 email, Wadim Zudilin gave a rebuttal to v1 of Kim's 2011 preprint: "The mistake (unfixable) is on p. 6, line after eq. (3.3). 'Without loss of generality' can be shown to work only for a finite set of n_k's; as the n_k are sufficiently large (and N is fixed), the inequality for epsilon is false." In a May 2013 email, Zudilin extended his rebuttal to cover v2, concluding that Kim's argument "implies that at least one of zeta(2), zeta(3), zeta(4) and zeta(5) is irrational, which is trivial." - Jonathan Sondow, May 06 2013

General: zeta(2*s + 1) = (A000364(s)/A331839(s)) * Pi^(2*s + 1) * Product_{k >= 1} (A002145(k)^(2*s + 1) + 1)/(A002145(k)^(2*s + 1) - 1), for s >= 1. - Dimitris Valianatos, Apr 27 2020

"6/Pi^2 is the probability that two randomly selected numbers will be coprime and also the probability that a randomly selected integer is 'squarefree.'" [Hardy and Wright] - C. Pickover.

In fact, the probability that any k randomly selected numbers will be coprimes is 1/Sum_{n>=1} n^(-k) = 1/zeta(k). - Robert G. Wilson v [corrected by Ilya Gutkovskiy, Aug 18 2018]

6/Pi^2 is also the diameter of a circle whose circumference equals the ratio of volume of a cuboid to the inscribed ellipsoid. 6/Pi^2 is also the diameter of a circle whose circumference equals the ratio of surface area of a cube to the inscribed sphere. - Omar E. Pol, Oct 08 2011

6/(Pi^2 * n^2) is the probability that two randomly selected positive integers will have a greatest common divisor equal to n, n >= 1. - Geoffrey Critzer, May 28 2013

Equals lim_{n->oo} (Sum_{k=1..n} phi(k)/k)/n, i.e., the limit mean value of phi(k)/k, where phi(k) is Euler's totient function. Proof is trivial using the formula for Sum_{k=1..n} phi(k)/k listed at the Wikipedia link. For the limit mean value of k/phi(k), see A082695. - Stanislav Sykora, Nov 14 2014

This is the probability that a random point on a square lattice is visible from the origin, i.e., there is no other lattice point that lies on the line segment between this point and the origin. - Amiram Eldar, Jul 08 2020