A390494 Irregular triangle read by rows: T(n,k) is the sum of nonprime numbers in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

1, 1, 1, 0, 5, 1, 0, 7, 1, 0, 13, 1, 0, 9, 1, 10, 1, 0, 23, 1, 0, 1, 14, 1, 0, 15, 29, 1, 0, 34, 1, 0, 35, 1, 0, 0, 21, 1, 22, 1, 0, 55, 1, 0, 25, 1, 26, 1, 0, 9, 27, 47, 1, 0, 62, 1, 0, 61, 1, 0, 0, 33, 1, 34, 1, 0, 35, 86, 1, 0, 1, 38, 1, 0, 0, 39, 83, 1, 0, 84, 1, 0, 5, 66, 1, 24, 45, 1, 46, 1, 0, 119, 1, 0, 49

Offset

1,5

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) - A390495(n,k).

Example

  --------------------------------------------------------------------
  |  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,  0;         |  [1], [3];                   |     2     |
  |  4 |   5;             |  [1, 2, 4];                  |     1     |
  |  5 |   1,  0;         |  [1], [5];                   |     2     |
  |  6 |   7;             |  [1, 2, 3, 6];               |     1     |
  |  7 |   1,  0;         |  [1], [7];                   |     2     |
  |  8 |  13;             |  [1, 2, 4, 8];               |     1     |
  |  9 |   1,  0,  9;     |  [1], [3], [9];              |     3     |
  | 10 |   1, 10;         |  [1, 2], [5, 10];            |     2     |
  | 11 |   1,  0;         |  [1], [11];                  |     2     |
  | 12 |  23;             |  [1, 2, 3, 4, 6, 12];        |     1     |
  | 13 |   1,  0;         |  [1], [13];                  |     2     |
  | 14 |   1, 14;         |  [1, 2], [7, 14];            |     2     |
  | 15 |   1,  0, 15;     |  [1], [3, 5], [15];          |     3     |
  | 16 |  29;             |  [1, 2, 4, 8, 16];           |     1     |
  | 17 |   1,  0;         |  [1], [17];                  |     2     |
  | 18 |  34;             |  [1, 2, 3, 6, 9, 18];        |     1     |
  | 19 |   1,  0;         |  [1], [19];                  |     2     |
  | 20 |  35;             |  [1, 2, 4, 5, 10, 20];       |     1     |
  | 21 |   1,  0,  0, 21; |  [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 nonprime numbers is 1 + 6 = 7, so T(78,1) = 7. In the second 2-dense sublist the sum of nonprime numbers is 26 + 39 + 78 = 143, so T(78,2) = 143.

Mathematica

A390494row[n_] := Map[Total[Select[#, !PrimeQ[#] &]] &, Split[Divisors[n], #2 <= 2*# &]];
Array[A390494row, 50] (* Paolo Xausa, Nov 18 2025 *)

Crossrefs

Row sums give A023890.

Cf. A018252, A027750, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A386604, A386989, A386992, A387030, A389592, A389761, A389762, A390443, A390495.

Sequence in context: A340841 A143148 A389454 * A198541 A200420 A176324

Adjacent sequences: A390491 A390492 A390493 * A390495 A390496 A390497

Keyword

nonn,tabf

Author

Omar E. Pol, Nov 11 2025

Status

approved