A386604 Irregular triangle read by rows: T(n,k) is the number of even numbers in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 0, 1, 1, 0, 0, 4, 0, 0, 1, 1, 0, 0, 0, 4, 0, 0, 3, 0, 0, 4, 0, 0, 0, 0, 1, 1, 0, 0, 6, 0, 0, 0, 1, 1, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 5, 0, 0, 0, 0, 1, 1, 0, 0, 0, 6, 0, 0, 1, 1, 0, 0, 0, 0, 6, 0, 0, 4, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0, 0, 8, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 2, 2, 0, 0, 4

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..12242 (rows 1..4000 of triangle, flattened).

Formula

T(n,k) = A384222(n,k) - A384225(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 |   1;           |  [1, 2];                     |     1     |
  |  3 |   0, 0;        |  [1], [3];                   |     2     |
  |  4 |   2;           |  [1, 2, 4];                  |     1     |
  |  5 |   0, 0;        |  [1], [5];                   |     2     |
  |  6 |   2;           |  [1, 2, 3, 6];               |     1     |
  |  7 |   0, 0;        |  [1], [7];                   |     2     |
  |  8 |   3;           |  [1, 2, 4, 8];               |     1     |
  |  9 |   0, 0, 0;     |  [1], [3], [9];              |     3     |
  | 10 |   1, 1;        |  [1, 2], [5, 10];            |     2     |
  | 11 |   0, 0;        |  [1], [11];                  |     2     |
  | 12 |   4;           |  [1, 2, 3, 4, 6, 12];        |     1     |
  | 13 |   0, 0;        |  [1], [13];                  |     2     |
  | 14 |   1, 1;        |  [1, 2], [7, 14];            |     2     |
  | 15 |   0, 0, 0;     |  [1], [3, 5], [15];          |     3     |
  | 16 |   4;           |  [1, 2, 4, 8, 16];           |     1     |
  | 17 |   0, 0;        |  [1], [17];                  |     2     |
  | 18 |   3;           |  [1, 2, 3, 6, 9, 18];        |     1     |
  | 19 |   0, 0;        |  [1], [19];                  |     2     |
  | 20 |   4;           |  [1, 2, 4, 5, 10, 20];       |     1     |
  | 21 |   0, 0, 0, 0;  |  [1], [3], [7], [21];        |     4     |
   ...
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] and [5, 10]. Each sublist has only one even number, so the row 10 is [1, 1].

Mathematica

A386604row[n_] := Map[Count[#, _?EvenQ] &, Split[Divisors[n], #2 <= 2*# &]];
Array[A386604row, 50] // Flatten (* Paolo Xausa, Oct 28 2025 *)

Crossrefs

Row sums give A183063.

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

Sequence in context: A394202 A033719 A171608 * A382487 A307985 A024164

Adjacent sequences: A386601 A386602 A386603 * A386605 A386606 A386607

Keyword

nonn,tabf

Author

Omar E. Pol, Oct 22 2025

Status

approved