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)
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)
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.
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
Conjecture 4:
A005279 are the numbers i such that
A001227(i) is greater than the number of 2-dense sublists of divisors of i. -
Omar E. Pol, Feb 22 2026
