A384222 Irregular triangle read by rows: T(n,k) is the length of the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

1, 2, 1, 1, 3, 1, 1, 4, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 6, 1, 1, 2, 2, 1, 2, 1, 5, 1, 1, 6, 1, 1, 6, 1, 1, 1, 1, 2, 2, 1, 1, 8, 1, 1, 1, 2, 2, 1, 1, 1, 1, 6, 1, 1, 8, 1, 1, 6, 1, 1, 1, 1, 2, 2, 1, 2, 1, 9, 1, 1, 2, 2, 1, 1, 1, 1, 8, 1, 1, 8, 1, 1, 3, 3, 1, 4, 1, 2, 2, 1, 1, 10, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 3, 1, 1, 8

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.

Row n has only one term, which is A000005(n), if and only if n is in A174973.

Conjecture 1: row n is a palindromic composition of A000005(n).

If the conjecture is true then this triangle should be a companion of A237270 in the sense that here the n-th row should be a palindromic composition of sigma_0(n) = A000005(n) and the n-th row of A237270 is a palindromic composition of sigma_1(n) = A000203(n).

A384149(n,k) is the sum of the terms in the k-th sublist of divisors of n. In the comments of A384149 it is conjectured that the row lengths of that triangle give A237271. If that conjecture is true then here the row lengths should also be A237271 and therefore A237271(n) could be defined also as the number of 2-dense sublists of divisors of n.

Links

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

Formula

From Omar E. Pol, Oct 22 2025: (Start)

T(n,k) = A384225(n,k) + A386604(n,k).

T(n,k) = A387030(n,k) + A386992(n,k).

T(n,k) = A389761(n,k) + A389762(n,k). (End)

T(n,k) = A384225(n,k) * A001511(n) = A384225(n,k) * A000005(n) / A001227(n). (conjectured). - Omar E. Pol, Apr 13 2026

Example

  ----------------------------------------------------------------
  |  n | Row n of     |  List of divisors of n       | Number of |
  |    | the triangle |  [with sublists in brackets] | sublists  |
  ----------------------------------------------------------------
  |  1 |  1;          |  [1];                        |     1     |
  |  2 |  2;          |  [1, 2];                     |     1     |
  |  3 |  1, 1;       |  [1], [3];                   |     2     |
  |  4 |  3;          |  [1, 2, 4];                  |     1     |
  |  5 |  1, 1;       |  [1], [5];                   |     2     |
  |  6 |  4;          |  [1, 2, 3, 6];               |     1     |
  |  7 |  1, 1;       |  [1], [7];                   |     2     |
  |  8 |  4;          |  [1, 2, 4, 8];               |     1     |
  |  9 |  1, 1, 1;    |  [1], [3], [9];              |     3     |
  | 10 |  2, 2;       |  [1, 2], [5, 10];            |     2     |
  | 11 |  1, 1;       |  [1], [11];                  |     2     |
  | 12 |  6;          |  [1, 2, 3, 4, 6, 12];        |     1     |
  | 13 |  1, 1;       |  [1], [13];                  |     2     |
  | 14 |  2, 2;       |  [1, 2], [7, 14];            |     2     |
  | 15 |  1, 2, 1;    |  [1], [3, 5], [15];          |     3     |
  | 16 |  5;          |  [1, 2, 4, 8, 16];           |     1     |
  | 17 |  1, 1;       |  [1], [17];                  |     2     |
  | 18 |  6;          |  [1, 2, 3, 6, 9, 18];        |     1     |
  | 19 |  1, 1;       |  [1], [19];                  |     2     |
  | 20 |  6;          |  [1, 2, 4, 5, 10, 20];       |     1     |
  | 21 |  1, 1, 1, 1; |  [1], [3], [7], [21];        |     4     |
  | 22 |  2, 2;       |  [1, 2], [11, 22];           |     2     |
  | 23 |  1, 1;       |  [1], [23];                  |     2     |
  | 24 |  8;          |  [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 sublists of divisors of 10 whose terms increase by a factor of at most 2, they are [1, 2] and [5, 10]. Each sublist has two terms, so the row 10 is [2, 2].
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 number of terms in the sublists are [1, 2, 1] respectively, so the row 15 is [1, 2, 1].
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 sublists of divisors of 78 whose terms increase by a factor of at most 2, they are [1, 2, 3, 6] and [13, 26, 39, 78]. The number of terms in the sublists are [4, 4] respectively, so the row 78 is [4, 4].
From Omar E. Pol, Jul 23 2025: (Start)
A visualization with symmetries of the list of divisors of the first 24 positive integers and the sublists of divisors is as shown below:
  ---------------------------------------------------------------------------------
  |     n     |                 List of divisors of n                 | Number of |
  |           |        [with sublists of divisors in brackets]        | sublists  |
  ---------------------------------------------------------------------------------
  |     1     |                          [1]                          |     1     |
  |     2     |                         [1 2]                         |     1     |
  |     3     |                        [1] [3]                        |     2     |
  |     4     |                       [1  2  4]                       |     1     |
  |     5     |                      [1]     [5]                      |     2     |
  |     6     |                     [1   2 3   6]                     |     1     |
  |     7     |                    [1]         [7]                    |     2     |
  |     8     |                   [1    2   4    8]                   |     1     |
  |     9     |                  [1]     [3]     [9]                  |     3     |
  |    10     |                 [1     2]   [5    10]                 |     2     |
  |    11     |                [1]                [11]                |     2     |
  |    12     |               [1      2  3 4  6     12]               |     1     |
  |    13     |              [1]                    [13]              |     2     |
  |    14     |             [1       2]       [7      14]             |     2     |
  |    15     |            [1]         [3   5]        [15]            |     3     |
  |    16     |           [1        2     4     8       16]           |     1     |
  |    17     |          [1]                            [17]          |     2     |
  |    18     |         [1         2   3     6   9        18]         |     1     |
  |    19     |        [1]                                [19]        |     2     |
  |    20     |       [1          2      4 5     10         20]       |     1     |
  |    21     |      [1]             [3]     [7]            [21]      |     4     |
  |    22     |     [1           2]              [11          22]     |     2     |
  |    23     |    [1]                                        [23]    |     2     |
  |    24     |   [1            2    3  4   6  8   12           24]   |     1     |
       ...
A similar structure show the positive integers in the square array A385000. (End)

Mathematica

A384222row[n_] := Map[Length, Split[Divisors[n], #2/# <= 2 &]];
Array[A384222row, 50] (* Paolo Xausa, Jul 08 2025 *)

Crossrefs

Row sums give A000005.

Cf. A000203, A005153, A027750, A174973 (2-dense numbers), A237271, A379288, A384149, A384225, A384226, A384928, A384930, A384931, A385000, A386604, A386984, A386989, A386992, A387030, A389592, A389761, A389762.

Cf. A001511, A001227.

Sequence in context: A210873 A224838 A030272 * A157128 A359899 A301376

Adjacent sequences: A384219 A384220 A384221 * A384223 A384224 A384225

Keyword

nonn,tabf

Author

Omar E. Pol, Jun 03 2025

Status

approved