A384226 Irregular triangle read by rows: T(n,k) is the sum of odd divisors in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

1, 1, 1, 3, 1, 1, 5, 4, 1, 7, 1, 1, 3, 9, 1, 5, 1, 11, 4, 1, 13, 1, 7, 1, 8, 15, 1, 1, 17, 13, 1, 19, 6, 1, 3, 7, 21, 1, 11, 1, 23, 4, 1, 5, 25, 1, 13, 1, 3, 9, 27, 8, 1, 29, 24, 1, 31, 1, 1, 3, 11, 33, 1, 17, 1, 12, 35, 13, 1, 37, 1, 19, 1, 3, 13, 39, 6, 1, 41, 32, 1, 43, 1, 11, 1, 32, 45, 1, 23, 1, 47, 4

Offset: 1

Author

Omar E. Pol, Jun 24 2025

Keywords

• nonn
• tabf

Comments

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.

Examples

			  --------------------------------------------------------------------
  |  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 |   4;             |  [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 |   4;             |  [1, 2, 3, 4, 6, 12];        |     1     |
  | 13 |   1, 13;         |  [1], [13];                  |     2     |
  | 14 |   1,  7;         |  [1, 2], [7, 14];            |     2     |
  | 15 |   1,  8, 15;     |  [1], [3, 5], [15];          |     3     |
  | 16 |   1;             |  [1, 2, 4, 8, 16];           |     1     |
  | 17 |   1, 17;         |  [1], [17];                  |     2     |
  | 18 |  13;             |  [1, 2, 3, 6, 9, 18];        |     1     |
  | 19 |   1, 19;         |  [1], [19];                  |     2     |
  | 20 |   6;             |  [1, 2, 4, 5, 10, 20];       |     1     |
  | 21 |   1,  3,  7, 21; |  [1], [3], [7], [21];        |     4     |
   ...
For n = 14 the list of divisors of 14 is [1, 2, 7, 14]. There are two sublists of divisors of 14 whose terms increase by a factor of at most 2, they are [1, 2] and [7, 14]. The sums of odd terms in the sublists are [1], [7] respectively, so the row 14 is [1, 7].
For n = 15 the list of divisors of 15 is [1, 3, 5, 15]. There are three sublists of divisors of 15 whose terms increase by a factor of at most 2, they are [1], [3, 5], [15]. The sums of terms in the sublists are [1, 8, 15] respectively, so the row 15 is [1, 8, 15].
For n = 16 the list of divisors of 16 is [1, 2, 4, 8, 16]. There is only one sublist of divisors of 16 whose terms increase by a factor of at most 2, that is the same as the list of divisors of 16, so the row 16 is [1].
For n = 2350 the list of divisors of 2350 is [1, 2, 5, 10, 25, 47, 50, 94, 235, 470, 1175, 2350]. There are five sublists of divisors of 2350 whose terms increase by a factor of at most 2, they are [1, 2], [5, 10], [25, 47, 50, 94], [235, 470], [1175, 2350]. The sums of odd terms in the sublists are [1, 5, 72, 235, 1175] respectively, so the row 2350 is [1, 5, 72, 235, 1175].
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10607 (rows 1..3500 of triangle, flattened).

Crossrefs

Row sums give A000593.

Cf. A174973 (2-dense numbers), A237271, A237593, A379288, A384149, A384222, A384225, A384928.

Programs

Mathematica

A384226row[n_] := Map[Total[Select[#, OddQ]] &, Split[Divisors[n], #2/# <= 2 &]];
Array[A384226row, 50] (* Paolo Xausa, Jul 08 2025 *)