A384930 Irregular triangle read by rows: T(n,k) is the sum of the terms of the (n-k+1)-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

1, 3, 3, 1, 7, 5, 1, 12, 7, 1, 15, 9, 3, 1, 15, 3, 11, 1, 28, 13, 1, 21, 3, 15, 8, 1, 31, 17, 1, 39, 19, 1, 42, 21, 7, 3, 1, 33, 3, 23, 1, 60, 25, 5, 1, 39, 3, 27, 9, 3, 1, 56, 29, 1, 72, 31, 1, 63, 33, 11, 3, 1, 51, 3, 35, 12, 1, 91, 37, 1, 57, 3, 39, 13, 3, 1, 90, 41, 1, 96, 43, 1, 77, 7, 45, 32, 1

Offset: 1

Author

Omar E. Pol, Jul 19 2025

Keywords

• nonn
• tabf
• changed

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 coincide with A237271.

Note that if the conjectures related to the 2-dense sublists of divisors of n are true so we have that essentially all sequences where the words "part" or "parts" are mentioned having cf. A237593 are also related to the 2-dense sublists of divisors of n, for example the square array A240062.

Examples

			  ---------------------------------------------------------------------
  |  n |   Row n of       |  List of divisors of n        | Number of |
  |    |   the triangle   |  [with sublists in brackets]  | sublists  |
  ---------------------------------------------------------------------
  |  1 |    1;            |  [1];                         |     1     |
  |  2 |    3;            |  [1, 2];                      |     1     |
  |  3 |    3, 1;         |  [1], [3];                    |     2     |
  |  4 |    7;            |  [1, 2, 4];                   |     1     |
  |  5 |    5, 1;         |  [1], [5];                    |     2     |
  |  6 |   12;            |  [1, 2, 3, 6];                |     1     |
  |  7 |    7, 1;         |  [1], [7];                    |     2     |
  |  8 |   15;            |  [1, 2, 4, 8];                |     1     |
  |  9 |    9, 3, 1;      |  [1], [3], [9];               |     3     |
  | 10 |   15  3;         |  [1, 2], [5, 10];             |     2     |
  | 11 |   11, 1;         |  [1], [11];                   |     2     |
  | 12 |   28;            |  [1, 2, 3, 4, 6, 12];         |     1     |
  | 13 |   13, 1;         |  [1], [13];                   |     2     |
  | 14 |   21, 3;         |  [1, 2], [7, 14];             |     2     |
  | 15 |   15, 8, 1;      |  [1], [3, 5], [15];           |     3     |
  | 16 |   31;            |  [1, 2, 4, 8, 16];            |     1     |
  | 17 |   17, 1;         |  [1], [17];                   |     2     |
  | 18 |   39;            |  [1, 2, 3, 6, 9, 18];         |     1     |
  | 19 |   19, 1;         |  [1], [19];                   |     2     |
  | 20 |   42;            |  [1, 2, 4, 5, 10, 20];        |     1     |
  | 21 |   21, 7, 3, 1;   |  [1], [3], [7], [21];         |     4     |
  | 22 |   33, 3;         |  [1, 2], [11, 22];            |     2     |
  | 23 |   23, 1;         |  [1], [23];                   |     2     |
  | 24 |   60;            |  [1, 2, 3, 4, 6, 8, 12, 24];  |     1     |
   ...
A conjectured relationship between a palindromic composition of sigma_0(n) = A000005(n) as n-th row of A384222 and the list of divisors of n as the n-th row of A027750 and a palindromic composition of sigma_1(n) = A000203(n) as the n-th row of A237270 and the diagram called "symmetric representation of sigma(n)" is as shown below with two examples.
.
For n = 10 the conjectured relationship is:
  10th row of A384222.......................: [   2  ], [   2  ]
  10th row of A027750.......................:   1, 2,     5, 10
  10th row of A027750 with sublists.........: [ 1, 2 ], [ 5, 10]
  10th row of A384149.......................: [   3  ], [  15  ]
  10th row of this triangle.................: [  15  ], [   3  ]
  10th row of the virtual sequence 2*A237270: [  18  ], [  18  ]
  10th row of A237270.......................: [   9  ], [   9  ]
.
The symmetric representation of sigma_1(10) in the first quadrant is as follows:
.
   _ _ _ _ _ _ 9
  |_ _ _ _ _  |
            | |_
            |_ _|_
                | |_ _  9
                |_ _  |
                    | |
                    | |
                    | |
                    | |
                    |_|
.
The diagram has two parts (or polygons) of areas  [9], [9] respectively, so the 10th row of A237270 is [9], [9] and sigma_1(10) = A000203(10) = 18.
.
For n = 15 the conjectured relationship is:
  15th row of A384222.......................: [ 1], [  2  ], [ 1]
  15th row of A027750.......................:   1,    3, 5,   15
  15th row of A027750 with sublists.........: [ 1], [ 3, 5], [15]
  15th row of A384149.......................: [ 1], [  8  ], [15]
  15th row of this triangle.................: [15], [  8  ], [ 1]
  15th row of the virtual sequence 2*A237270: [16], [ 16  ], [16]
  15th row of A237270.......................: [ 8], [  8  ], [ 8]
.
The symmetric representation of sigma_1(15) in the first quadrant is as follows:
.
   _ _ _ _ _ _ _ _ 8
  |_ _ _ _ _ _ _ _|
                  |
                  |_ _
                  |_  |_ 8
                    |   |_
                    |_ _  |
                        |_|_ _ _ 8
                              | |
                              | |
                              | |
                              | |
                              | |
                              | |
                              | |
                              |_|
.
The diagram has three parts (or polygons) of areas [8], [8], [8] respectively, so the 15th row of A237270 is [8], [8], [8] and sigma_1(15) = A000203(15) = 24.
.
For the relationship with Dyck paths, partitions of n into consecutive parts and odd divisors of n see A237593, A235791, A237591 and A379630.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..12242 (rows 1..4000 of triangle, flattened).

Crossrefs

Mirror of A384149.

Row sums give A000203.

Cf. A000105, A174973, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A240062, A245092, A262626, A379288, A379630, A384222, A384225, A384226, A384227.

Programs

Mathematica

A384930row[n_] := Reverse[Total[Split[Divisors[n], #2 <= 2*# &], {2}]];
Array[A384930row, 50] (* Paolo Xausa, Aug 14 2025 *)

Formula

T(n,k) = A384149(n,m+1-k), n >= 1, k >= 1, and m is the length of row n.

T(n,k) = 2*A237270(n,k) - A384149(n,k), n >= 1, k >= 1, (conjectured).