A390443 Irregular triangle read by rows: T(n,k) is the sum of even numbers in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

0, 2, 0, 0, 6, 0, 0, 8, 0, 0, 14, 0, 0, 0, 2, 10, 0, 0, 24, 0, 0, 2, 14, 0, 0, 0, 30, 0, 0, 26, 0, 0, 36, 0, 0, 0, 0, 2, 22, 0, 0, 56, 0, 0, 0, 2, 26, 0, 0, 0, 0, 48, 0, 0, 48, 0, 0, 62, 0, 0, 0, 0, 2, 34, 0, 0, 0, 78, 0, 0, 2, 38, 0, 0, 0, 0, 84, 0, 0, 64, 0, 0, 6, 66, 0, 0, 0, 2, 46, 0, 0, 120, 0, 0, 0, 2, 10, 50

Offset

1,2

Comments

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 give A237271.

Links

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

Formula

T(n,k) = A384149(n,k) - A384226(n,k).

Example

  --------------------------------------------------------------------
  |  n |  Row n of        |  List of divisors of n       | Number of |
  |    |  the triangle    |  [with sublists in brackets] | sublists  |
  --------------------------------------------------------------------
  |  1 |   0;             |  [1];                        |     1     |
  |  2 |   2;             |  [1, 2];                     |     1     |
  |  3 |   0,  0;         |  [1], [3];                   |     2     |
  |  4 |   6;             |  [1, 2, 4];                  |     1     |
  |  5 |   0,  0;         |  [1], [5];                   |     2     |
  |  6 |   8;             |  [1, 2, 3, 6];               |     1     |
  |  7 |   0,  0;         |  [1], [7];                   |     2     |
  |  8 |  14;             |  [1, 2, 4, 8];               |     1     |
  |  9 |   0,  0,  0;     |  [1], [3], [9];              |     3     |
  | 10 |   2, 10;         |  [1, 2], [5, 10];            |     2     |
  | 11 |   0,  0;         |  [1], [11];                  |     2     |
  | 12 |  24;             |  [1, 2, 3, 4, 6, 12];        |     1     |
  | 13 |   0,  0;         |  [1], [13];                  |     2     |
  | 14 |   2, 14;         |  [1, 2], [7, 14];            |     2     |
  | 15 |   0,  0,  0;     |  [1], [3, 5], [15];          |     3     |
  | 16 |  30;             |  [1, 2, 4, 8, 16];           |     1     |
  | 17 |   0,  0;         |  [1], [17];                  |     2     |
  | 18 |  26;             |  [1, 2, 3, 6, 9, 18];        |     1     |
  | 19 |   0,  0;         |  [1], [19];                  |     2     |
  | 20 |  36;             |  [1, 2, 4, 5, 10, 20];       |     1     |
  | 21 |   0,  0,  0,  0; |  [1], [3], [7], [21];        |     4     |
   ...
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]. In the first 2-dense sublist the sum of even numbers is 2 + 6 = 8, so T(78,1) = 8. In the second 2-dense sublist the sum of even numbers is 26 + 78 = 104, so T(78,2) = 104.

Mathematica

A390443row[n_] := Map[Total[Select[#, EvenQ]] &, Split[Divisors[n], #2 <= 2*# &]];
Array[A390443row, 50] (* Paolo Xausa, Nov 11 2025 *)

Crossrefs

Row sums give A146076.

Cf. A027750, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A386604, A386989, A386992, A387030, A389592, A389761, A389762.

Sequence in context: A259857 A364790 A094785 * A265856 A389592 A035536

Adjacent sequences: A390440 A390441 A390442 * A390444 A390445 A390446

Keyword

nonn,tabf

Author

Omar E. Pol, Nov 05 2025

Status

approved