cp's OEIS Frontend

Also (1) number of ways to write n as difference of two triangular numbers (A000217), see A136107; (2) number of ways to arrange n identical objects in a trapezoid. - Tom Verhoeff

Also number of partitions of n into consecutive positive integers including the trivial partition of length 1 (e.g., 9 = 2+3+4 or 4+5 or 9 so a(9)=3). (Useful for cribbage players.) See A069283. - Henry Bottomley, Apr 13 2000

This has been described as Sylvester's theorem, but to reduce ambiguity I suggest calling it Sylvester's enumeration. - Gus Wiseman, Oct 04 2022

a(n) is also the number of factors in the factorization of the Chebyshev polynomial of the first kind T_n(x). - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003

Number of factors in the factorization of the polynomial x^n+1 over the integers. See also A000005. - T. D. Noe, Apr 16 2003

a(n) = 1 if and only if n is a power of 2 (see A000079). - Lekraj Beedassy, Apr 12 2005

Number of occurrences of n in A049777. - Philippe Deléham, Jun 19 2005

For n odd, n is prime if and only if a(n) = 2. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Sep 10 2005

Also number of partitions of n such that if k is the largest part, then each of the parts 1,2,...,k-1 occurs exactly once. Example: a(9)=3 because we have [3,3,2,1],[2,2,2,2,1] and [1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 07 2006

Also the number of factors of the n-th Lucas polynomial. - T. D. Noe, Mar 09 2006

Lengths of rows of triangle A182469;

Denoted by Delta_0(n) in Glaisher 1907. - Michael Somos, May 17 2013

Also the number of partitions p of n into distinct parts such that max(p) - min(p) < length(p). - Clark Kimberling, Apr 18 2014

Row sums of triangle A247795. - Reinhard Zumkeller, Sep 28 2014

Row sums of triangle A237048. - Omar E. Pol, Oct 24 2014

A069288(n) <= a(n). - Reinhard Zumkeller, Apr 05 2015

A000203, A000593 and this sequence have the same parity: A053866. - Omar E. Pol, May 14 2016

a(n) is equal to the number of ways to write 2*n-1 as (4*x + 2)*y + 4*x + 1 where x and y are nonnegative integers. Also a(n) is equal to the number of distinct values of k such that k/(2*n-1) + k divides (k/(2*n-1))^(k/(2*n-1)) + k, (k/(2*n-1))^k + k/(2*n-1) and k^(k/(2*n-1)) + k/(2*n-1). - Juri-Stepan Gerasimov, May 23 2016, Jul 15 2016

Also the number of odd divisors of n*2^m for m >= 0. - Juri-Stepan Gerasimov, Jul 15 2016

a(n) is odd if and only if n is a square or twice a square. - Juri-Stepan Gerasimov, Jul 17 2016

a(n) is also the number of subparts in the symmetric representation of sigma(n). For more information see A279387 and A237593. - Omar E. Pol, Nov 05 2016

a(n) is also the number of partitions of n into an odd number of equal parts. - Omar E. Pol, May 14 2017 [This follows from the g.f. Sum_{k >= 1} x^k/(1-x^(2*k)). - N. J. A. Sloane, Dec 03 2020]

The alternating sum of the squares of the elements of the n-th row equals the sum of all divisors of all positive integers <= n, i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*(T(n,k))^2 = A024916(n).

Row n has length A003056(n) hence the first element of column k is in row A000217(k).

For more information see A236104.

The sum of row n gives A060831(n), the sum of the number of odd divisors of all positive integers <= n. - Omar E. Pol, Mar 01 2014. [An equivalent assertion is that the sum of row n of A237048 is the number of odd divisors of n, and this was proved by Hartmut F. W. Hoft in a comment in A237048. - N. J. A. Sloane, Dec 07 2020]

Comments from Franklin T. Adams-Watters on sequences related to the "symmetric representation of sigma" in A235791 and related sequences, Mar 31 2014: (Start)

The place to start is with A235791, which is very simple. Then go to A237591, also very simple, and A237593, still very simple.

You then need to interpret the rows of A237593 as Dyck paths. This interpretation is in terms of run lengths, so 2,1,1,2 means up twice, down once, up once, and down twice. Because the rows of A237593 are symmetric and of even length, this path will always be symmetric.

Now the surprising fact is that the areas enclosed by the Dyck path for n (laid on its side) always includes the area enclosed for n-1; and the number of squares added is sigma(n).

Finally, look at the connected areas enclosed by n but not by n-1; the size of these areas is the symmetric representation of sigma. (End)

From Hartmut F. W. Hoft, Apr 07 2014: (Start)

Mathematica function has been written to check the first property up to n = 20000.

T(n,(sqrt(8n+1)-1)/2+1) = 0 for all n >= 1, which is useful for formulas for A237591 and A237593. (End)

Alternating row sums give A240542. - Omar E. Pol, Apr 16 2014

Conjecture: T(n,k) is also the total number of partitions of all positive integers <= n into exactly k consecutive parts, i.e., the partial column sum of A285898, or in accordance with the triangles of the same family: the partial column sum of A237048. - Omar E. Pol, Apr 28 2017, Nov 24 2020

The above conjecture is true. The proof will be added soon (it uses the generating function for the columns). - N. J. A. Sloane, Nov 24 2020

T(n,k) is also the total length of all line segments between the k-th vertex and the central vertex of the largest Dyck path of the symmetric representation of sigma(n). In other words: T(n,k) is the sum of the last (A003056(n)-k+1) terms of the n-th row of A237591. - Omar E. Pol, Sep 07 2021

T(n,k) is also the Manhattan distance between the k-th vertex and the central vertex of the Dyck path described in the n-th row of the triangle A237593. - Omar E. Pol, Jan 11 2023

The diagram of the symmetry of sigma has been via A196020 --> A236104 --> A235791 --> A237591 --> A237593.

For more information see A237270.

a(n) is also the number of terraces at n-th level (starting from the top) of the stepped pyramid described in A245092. - Omar E. Pol, Apr 20 2016

a(n) is also the number of subparts in the first layer of the symmetric representation of sigma(n). For the definion of "subpart" see A279387. - Omar E. Pol, Dec 08 2016

Note that the number of subparts in the symmetric representation of sigma(n) equals A001227(n), the number of odd divisors of n. (See the second example). - Omar E. Pol, Dec 20 2016

From Hartmut F. W. Hoft, Dec 26 2016: (Start)

Using odd prime number 3, observe that the 1's in the 3^k-th row of the irregular triangle of A237048 are at index positions

3^0 < 2*3^0 < 3^1 < 2*3^1 < ... < 2*3^((k-1)/2) < 3^(k/2) < ...

the last being 2*3^((k-1)/2) when k is odd and 3^(k/2) when k is even. Since odd and even index positions alternate, each pair (3^i, 2*3^i) specifies one part in the symmetric representation with a center part present when k is even. A straightforward count establishes that the symmetric representation of 3^k, k>=0, has k+1 parts. Since this argument is valid for any odd prime, every positive integer occurs infinitely many times in the sequence. (End)

a(n) = number of runs of consecutive nonzero terms in row n of A262045. - N. J. A. Sloane, Jan 18 2021

Indices of odd terms give A071562. Indices of even terms give A071561. - Omar E. Pol, Feb 01 2021

a(n) is also the number of prisms in the three-dimensional version of the symmetric representation of k*sigma(n) where k is the height of the prisms, with k >= 1. - Omar E. Pol, Jul 01 2021

With a(1) = 0; a(n) is also the number of parts in the symmetric representation of A001065(n), the sum of aliquot parts of n. - Omar E. Pol, Aug 04 2021

The parity of this sequence is also the characteristic function of numbers that have middle divisors. - Omar E. Pol, Sep 30 2021

a(n) is also the number of polycubes in the 3D-version of the ziggurat of order n described in A347186. - Omar E. Pol, Jun 11 2024

Conjecture 1: a(n) is the number of odd divisors of n except the "e" odd divisors described in A005279. Thus a(n) is the length of the n-th row of A379288. - Omar E. Pol, Dec 21 2024

The conjecture 1 was checked up n = 10000 by Amiram Eldar. - Omar E. Pol, Dec 22 2024

The conjecture 1 is true. For a proof see A379288. - Hartmut F. W. Hoft, Jan 21 2025

From Omar E. Pol, Jul 31 2025: (Start)

Conjecture 2: a(n) is the number of 2-dense sublists of divisors of n.

We call "2-dense sublists of divisors of n" to the maximal sublists of divisors of n whose terms increase by a factor of at most 2.

In a 2-dense sublist of divisors of n the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of n.

Example: for n = 10 the list of divisors of 10 is [1, 2, 5, 10]. There are two 2-dense sublists of divisors of 10, they are [1, 2], [5, 10], so a(10) = 2.

The conjecture 2 is essentially the same as the second conjecture in the Comments of A384149. See also Peter Munn's formula in A237270.

The indices where a(n) = 1 give A174973 (2-dense numbers). See the proof there. (End)

Conjecture 3: a(n) is the number of divisors p of n such that p is greater than twice the adjacent previous divisor of n. The divisors p give the n-th row of A379288. - Omar E. Pol, Aug 02 2025

This is a triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists successive blocks of k consecutive terms, where the m-th block starts with m, m>=1, and the first element of column k is in row k*(k+1)/2.

The partitions of n into consecutive parts are represented from the row n up to row A288529(n) as maximum, but in increasing order, exclusively in the columns where the blocks begin.

More precisely, the partition of n into exactly k consecutive parts (if such partition exists) is represented in the column k from the row n up to row n + k - 1 (see examples).

A288772(n) is the minimum number of rows that are required to represent in this table the partitions of all positive integers <= n into consecutive parts.

A288773(n) is the largest of all positive integers whose partitions into consecutive parts can be totally represented in the first n rows of this table.

A288774(n) is the largest positive integers whose partitions into consecutive parts can be totally represented in the first n rows of this table.

For a theorem related to this table see A286000.

Conjecture: row n is formed by the odd-indexed terms of the n-th row of triangle A280850 together with the even-indexed terms of the same row but listed in reverse order. Examples: the 15th row of A280850 is [8, 8, 7, 0, 1] so the 15th row of this triangle is [8, 7, 1, 0, 8]. The 75th row of A280850 is [38, 38, 21, 0, 3, 3, 0, 0, 0, 21, 0] so the 75h row of this triangle is [38, 21, 3, 0, 0, 0, 21, 0, 3, 0, 38].

For the definition of "subparts" see A279387.

For more information about the mentioned Dyck paths see A237593.

T(n,k) could be called the "charge" of the k-th peak of the largest Dyck path of the symmetric representation of sigma(n).

The number of zeros in row n is A238005(n). - Omar E. Pol, Sep 11 2021

This sequence is a companion to A279387 in which each contiguous sequence of identical widths w in A249223 are replaced by a single entry of w. Using the resulting distribution pattern of widths across all parts of the symmetric representation of sigma(n) the subparts at each level are counted in A279387.

The sequence of widths are computed first to the diagonal of the symmetric representation of sigma only for those numbers in set F defined in A341971. Then the reversed list less its first number is appended so that the width at the diagonal is not listed twice. Thus every row contains an odd number of entries and is symmetric about its center entry.

Let 1 <= n, 1 <= d <= s = A001227(n) and 1 <= k <= r = floor((sqrt(8*n + 1) - 1)/2). Let Q(n,d) be row n in the triangle of A341970, R(n,d) be row n in the triangle of A341970 and S(n,d) = R(n,Q(n,d)), then T(n,e) = S(n,e) for 1 <= e <= s and T(n,e) = S(n,2*s - e) for s < e <= 2*s - 1 is row n for this sequence.

Also sum of all odd-indexed parts minus the sum of all even-indexed parts of all partitions of n (Cf. A206563). - Omar E. Pol, Feb 12 2012

Column 1 of A206563. - Omar E. Pol, Feb 15 2012

Suppose that p=[p(1),p(2),p(3),...] is a partition of n with parts in nonincreasing order. Let f(p) = p(1) - p(2) + p(3) - ... be the alternating sum of parts of p and let F(n) = sum of alternating sums of all partitions of n. Conjecture: F(n) = A066897(n) for n >= 1. - Clark Kimberling, May 17 2019

From Omar E. Pol, Apr 02 2023: (Start)

Convolution of A000041 and A001227.

Convolution of A002865 and A060831.

a(n) is also the total number of odd divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned odd divisors are also all odd parts of all partitions of n. (End)

a(n) is odd iff n is a term of A067567 (proof: n*p(n) = the sum of the parts in all the partitions of n == the number of odd parts in all partitions of n (mod 2). Hence the number of odd parts in all partitions of n is odd iff n*p(n) is odd, equivalently, iff both n and p(n) are odd). - Peter Bala, Jan 11 2025

The total area (or total number of cells) of the diagram after n stages is equal to A024916(n), the sum of all divisors of all positive integers <= n.

Note that the region between the virtual circumscribed square and the diagram is a symmetric polygon whose area is equal to A004125(n), see example.

For more information see A237593 and A237270.

a(n) is also the total number of terraces of the stepped pyramid with n levels described in A245092. - Omar E. Pol, Apr 20 2016

Row n has length A001227(n), the number of odd divisors of n, and also the number of entries in row n of A341970.

Let 1 <= n, 1 <= d <= A001227(n) and k = A341970(A060831(n-1) + d). Expression s(n, k) = r*n - r*(r+1)*(r+2)/6 + k with r = floor((sqrt(8*n + 1) - 1)/2) translates position (row n, column k) in the triangle of A235791 to its position in sequence A235791.

The absolute values in row n are the smallest parts of the partitions of n into consecutive parts (with the partitions ordered by increasing number of parts). - Omar E. Pol, Dec 31 2024

The number of entries in the n-th row of the table of this sequence is A001227(n), the number of odd divisors of n.

Let number n = 2^s * q with s >= 0 and q odd, let row(n) = floor( (sqrt(8*n+1) - 1)/2 ), let D_n = { d : d odd divisor of n and d <= row(n) }, let E_n = { e : e = 2^(s+1) * d, d in D_n and e <= row(n) } and let F_n be the union of D_n and E_n with its elements listed in increasing order. Then the numbers in F_n are exactly the positions of 1's in row n of A237048 and the numbers in row n of this sequence.