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 coincide with A237271.
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..12242 (rows 1..4000 of triangle, flattened).
FORMULA
EXAMPLE
---------------------------------------------------------------------
| 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.
.
MATHEMATICA
A384930row[n_] := Reverse[Total[Split[Divisors[n], #2 <= 2*# &], {2}]];
Array[A384930row, 50] (* Paolo Xausa, Aug 14 2025 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Jul 19 2025
STATUS
approved