cp's OEIS Frontend

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

Conjecture 1: where records occur in A237271. - Omar E. Pol, Dec 27 2016

For more information about the symmetric representation of sigma see A237270, A237593.

This sequence of (first occurrence of) parts appears to be strictly increasing in contrast to sequence A250070 of (first occurrence of) maximum widths. - Hartmut F. W. Hoft, Dec 09 2014

Conjecture 2: all terms are odd numbers. - Omar E. Pol, Oct 14 2018

Proof of Conjecture 2: Let n = 2^m * q with m>0 and q odd; then the 1's in even positions of row n in the triangle of A237048 are at positions 2^(m+1) * d <= row(n) where d divides q. For n/2 the even positions of 1's occur at the smaller values 2^m * d <= row(n/2), thus either keeping or reducing widths (A249223) of parts in the symmetric representation of sigma for n/2 inherited from row n. Therefore the number of parts for n is at most as large as for n/2, i.e., all numbers in this sequence are odd. - Hartmut F. W. Hoft, Sep 22 2021

Observation: at least for n = 1..21 we have that 2*a(n) < a(n+1). - Omar E. Pol, Sep 22 2021

From Omar E. Pol, Jul 28 2025: (Start)

Conjecture 3: a(n) is the smallest number k having n 2-dense sublists of divisors of k.

The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.

In a sublist of divisors of k the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of k.

An example of the conjecture 3 for n = 1..5 is as shown below:

----------------------------------------------------

| | List of divisors of k | | |

| k | [with sublists in brackets] | n | a(n) |

----------------------------------------------------

| 1 | [1]; | 1 | 1 |

| 3 | [1], [3]; | 2 | 3 |

| 9 | [1], [3], [9]; | 3 | 9 |

| 21 | [1], [3], [7], [21]; | 4 | 21 |

| 63 | [1], [3], [7, 9], [21], [63]; | 5 | 63 |

(End)

Conjecture 4: a(n) is the smallest number k having n divisors p of k such that p is greater than twice the adjacent previous divisor of k. - Omar E. Pol, Aug 05 2025

The excluded divisors are the odd divisors e listed in A005279.

Conjecture 1: the row lengths are given by A237271 (true for at least the first 10000 terms of A237271)

From Hartmut F. W. Hoft, Jan 09 2025: (Start)

Proof of Conjecture 1:

An entire part of SRS(n), n = 2^k * q with k >= 0 and q odd, up to the diagonal is described in row n of A249223 by a 1 in position d, an odd divisor of n, 0's in positions d-1 and 2^(k+1) * f, f >= d an odd divisor of n, and nonzero numbers that increase or decrease by 1 in between.

The odd divisors e of n with d < e < 2^(k+1) * f are the "e" odd divisors of A005279 since for divisor s of n, d < s < e < 2*s < 2^(k+1) * f holds.

The odd divisors u of n greater than A003056(n) are encoded by the 2^(k+1) * f above as u = q/f and odd divisors d < A003056(n) are also encoded as 2^(k+1) * q/d. Then odd divisors e of n with q/f < e < 2^(k+1) * q/d are the "e" odd divisors of A005279 since for divisor t of n, q/f < t < e < 2*t < 2^(k+1) * q/d holds.

For a part containing the diagonal the inequalities above hold on the respective sides of the diagonal.

As a consequence the number of entries in row n of this triangle equals A237271(n). (End)

From Omar E. Pol, Jun 26 2025: (Start)

Conjecture 2: T(n,m) is the smallest number in the m-th 2-dense sublist 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.

If the conjecture is true so row sums give A379379 and the row lengths give A237271, and the same row lengths have the sequences A384222, A384225 and A384226. Also the conjecture of A384149 should be true.

Observation: at least for the first 5000 rows (the first 15542 terms) the conjecture 2 coincides with the definition from the Name section and the row lengths give A237271.

An example of the conjecture 2, for n = 1..24 is as shown below:

-------------------------------------------------------------------

| n | Row n of | List of divisors of n | Number of |

| | the triangle | [with sublists in brackets] | sublists |

--------------------------------------------------------------------

| 1 | 1; | [1]; | 1 |

| 2 | 1; | [1, 2]; | 1 |

| 3 | 1, 3; | [1], [3]; | 2 |

| 4 | 1; | [1, 2, 4]; | 1 |

| 5 | 1, 5; | [1], [5]; | 2 |

| 6 | 1; | [1, 2, 3, 6]; | 1 |

| 7 | 1, 7; | [1], [7]; | 2 |

| 8 | 1; | [1, 2, 4, 8]; | 1 |

| 9 | 1, 3, 9; | [1], [3], [9]; | 3 |

| 10 | 1, 5; | [1, 2], [5, 10]; | 2 |

| 11 | 1, 11; | [1], [11]; | 2 |

| 12 | 1; | [1, 2, 3, 4, 6, 12]; | 1 |

| 13 | 1, 13; | [1], [13]; | 2 |

| 14 | 1, 7; | [1, 2], [7, 14]; | 2 |

| 15 | 1, 3, 15; | [1], [3, 5], [15]; | 3 |

| 16 | 1; | [1, 2, 4, 8, 16]; | 1 |

| 17 | 1, 17; | [1], [17]; | 2 |

| 18 | 1; | [1, 2, 3, 6, 9, 18]; | 1 |

| 19 | 1, 19; | [1], [19]; | 2 |

| 20 | 1; | [1, 2, 4, 5, 10, 20]; | 1 |

| 21 | 1, 3, 7, 21; | [1], [3], [7], [21]; | 4 |

| 22 | 1, 11; | [1, 2], [11, 22]; | 2 |

| 23 | 1, 23; | [1], [23]; | 2 |

| 24 | 1; | [1, 2, 3, 4, 6, 8, 12, 24]; | 1 |

...

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]. The smallest numbers in the sublists are [1, 5] respectively, so the row 10 is [1, 5].

For n = 15 the list of divisors of 15 is [1, 3, 5, 15]. There are three 2-dense sublists of divisors of 15, they are [1], [3, 5], [15]. The smallest numbers in the sublists are [1, 3, 15] respectively, so the row 15 is [1, 3, 15].

78 is the first practical number A005153 not in A174973. For n = 78 the list of divisors of 78 is [1, 2, 3, 6, 13, 26, 39, 78]. There are two 2-dense sublists of divisors of 78, they are [1, 2, 3, 6] and [13, 26, 39, 78]. The smallest numbers in the sublists are [1, 13] respectively, so the row 78 is [1, 13].

(End)

Conjecture 3: T(n,m) is the m-th divisor p of n such that p is greater than twice the adjacent previous divisor of n. - Omar E. Pol, Aug 02 2025

T(n,k) is the number of odd numbers in the k-th sublist of divisors of n whose terms increase by a factor of at most 2,

In a 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.

At least for the first 1000 rows the row lengths give A237271.

Observation: at least the first 33 rows (or first 62 terms) coincide with A280940.

T(n,k) is the sum of odd numbers in the k-th sublist (or subsequence) of divisors of n such that the ratio of adjacent divisors in every sublist is at most 2.

In a 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.

It shares the odd-indexed rows with A384149.

At least for the first 1000 rows the row lengths give A237271.

In a 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.

The 2-dense sublists of divisors of n are the maximal sublists whose terms increase by a factor of at most 2.

At least for the first 1000 rows the row lengths coincide with A237271.

Note that if the conjectures related to the 2-dense sublists of divisors of n are true so we have that essentially all sequences where the words "part" or "parts" are mentioned having cf. A237593 are also related to the 2-dense sublists of divisors of n, for example the square array A240062.

By definition a(n) is also the number of 2-dense sublists of divisors of the n-th generalized hexagonal number.

In a sublist of divisors of k the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of k.

The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.

Conjecture: all odd indexed terms are odd.

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

The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.

In a sublist of divisors of k the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of k.

The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.

Conjecture: all terms are odd.

In a 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.

The 2-dense sublists of divisors of n are the maximal sublists whose terms increase by a factor of at most 2.

It is conjectured that row lengths are given by A237271.