A379288 -id:A379288 - cp's OEIS frontend

A237271 Number of parts in the symmetric representation of sigma(n).

1, 1, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 2, 3, 1, 2, 1, 2, 1, 4, 2, 2, 1, 3, 2, 4, 1, 2, 1, 2, 1, 4, 2, 3, 1, 2, 2, 4, 1, 2, 1, 2, 2, 3, 2, 2, 1, 3, 3, 4, 2, 2, 1, 4, 1, 4, 2, 2, 1, 2, 2, 5, 1, 4, 1, 2, 2, 4, 3, 2, 1, 2, 2, 4, 2, 3, 2, 2, 1, 5, 2, 2, 1, 4, 2, 4, 1, 2, 1

Offset: 1

Omar E. Pol, Feb 25 2014

nonn

The diagram of the symmetry of sigma has been via A196020 --> A236104 --> A235791 --> A237591 --> A237593.
For more information see A237270.
a(n) is also the number of terraces at n-th level (starting from the top) of the stepped pyramid described in A245092. - Omar E. Pol, Apr 20 2016
a(n) is also the number of subparts in the first layer of the symmetric representation of sigma(n). For the definion of "subpart" see A279387. - Omar E. Pol, Dec 08 2016
Note that the number of subparts in the symmetric representation of sigma(n) equals A001227(n), the number of odd divisors of n. (See the second example). - Omar E. Pol, Dec 20 2016
From Hartmut F. W. Hoft, Dec 26 2016: (Start)
Using odd prime number 3, observe that the 1's in the 3^k-th row of the irregular triangle of A237048 are at index positions
     3^0 < 2*3^0 < 3^1 < 2*3^1 < ... < 2*3^((k-1)/2) < 3^(k/2) < ...
  the last being 2*3^((k-1)/2) when k is odd and 3^(k/2) when k is even. Since odd and even index positions alternate, each pair (3^i, 2*3^i) specifies one part in the symmetric representation with a center part present when k is even. A straightforward count establishes that the symmetric representation of 3^k, k>=0, has k+1 parts. Since this argument is valid for any odd prime, every positive integer occurs infinitely many times in the sequence. (End)
a(n) = number of runs of consecutive nonzero terms in row n of A262045. - N. J. A. Sloane, Jan 18 2021
Indices of odd terms give A071562. Indices of even terms give A071561. - Omar E. Pol, Feb 01 2021
a(n) is also the number of prisms in the three-dimensional version of the symmetric representation of k*sigma(n) where k is the height of the prisms, with k >= 1. - Omar E. Pol, Jul 01 2021
With a(1) = 0; a(n) is also the number of parts in the symmetric representation of A001065(n), the sum of aliquot parts of n. - Omar E. Pol, Aug 04 2021
The parity of this sequence is also the characteristic function of numbers that have middle divisors. - Omar E. Pol, Sep 30 2021
a(n) is also the number of polycubes in the 3D-version of the ziggurat of order n described in A347186. - Omar E. Pol, Jun 11 2024
Conjecture 1: a(n) is the number of odd divisors of n except the "e" odd divisors described in A005279. Thus a(n) is the length of the n-th row of A379288. - Omar E. Pol, Dec 21 2024
The conjecture 1 was checked up n = 10000 by Amiram Eldar. - Omar E. Pol, Dec 22 2024
The conjecture 1 is true. For a proof see A379288. - Hartmut F. W. Hoft, Jan 21 2025
From Omar E. Pol, Jul 31 2025: (Start)
Conjecture 2: a(n) is the number of 2-dense sublists of divisors of n.
We call "2-dense sublists of divisors of n" to the maximal sublists of divisors of n whose terms increase by a factor of at most 2.
In a 2-dense 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.
Example: 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], [5, 10], so a(10) = 2.
The conjecture 2 is essentially the same as the second conjecture in the Comments of A384149. See also Peter Munn's formula in A237270.
The indices where a(n) = 1 give A174973 (2-dense numbers). See the proof there. (End)
Conjecture 3: a(n) is the number of divisors p of n such that p is greater than twice the adjacent previous divisor of n. The divisors p give the n-th row of A379288. - Omar E. Pol, Aug 02 2025

			Illustration of initial terms (n = 1..12):
---------------------------------------------------------
n   A000203  A237270    a(n)            Diagram
---------------------------------------------------------
.                               _ _ _ _ _ _ _ _ _ _ _ _
1       1      1         1     |_| | | | | | | | | | | |
2       3      3         1     |_ _|_| | | | | | | | | |
3       4      2+2       2     |_ _|  _|_| | | | | | | |
4       7      7         1     |_ _ _|    _|_| | | | | |
5       6      3+3       2     |_ _ _|  _|  _ _|_| | | |
6      12      12        1     |_ _ _ _|  _| |  _ _|_| |
7       8      4+4       2     |_ _ _ _| |_ _|_|    _ _|
8      15      15        1     |_ _ _ _ _|  _|     |
9      13      5+3+5     3     |_ _ _ _ _| |      _|
10     18      9+9       2     |_ _ _ _ _ _|  _ _|
11     12      6+6       2     |_ _ _ _ _ _| |
12     28      28        1     |_ _ _ _ _ _ _|
...
For n = 9 the sum of divisors of 9 is 1+3+9 = A000203(9) = 13. On the other hand the 9th set of symmetric regions of the diagram is formed by three regions (or parts) with 5, 3 and 5 cells, so the total number of cells is 5+3+5 = 13, equaling the sum of divisors of 9. There are three parts: [5, 3, 5], so a(9) = 3.
From _Omar E. Pol_, Dec 21 2016: (Start)
Illustration of the diagram of subparts (n = 1..12):
---------------------------------------------------------
n   A000203  A279391  A001227           Diagram
---------------------------------------------------------
.                               _ _ _ _ _ _ _ _ _ _ _ _
1       1      1         1     |_| | | | | | | | | | | |
2       3      3         1     |_ _|_| | | | | | | | | |
3       4      2+2       2     |_ _|  _|_| | | | | | | |
4       7      7         1     |_ _ _|  _ _|_| | | | | |
5       6      3+3       2     |_ _ _| |_|  _ _|_| | | |
6      12      11+1      2     |_ _ _ _|  _| |  _ _|_| |
7       8      4+4       2     |_ _ _ _| |_ _|_|  _ _ _|
8      15      15        1     |_ _ _ _ _|  _|  _| |
9      13      5+3+5     3     |_ _ _ _ _| |  _|  _|
10     18      9+9       2     |_ _ _ _ _ _| |_ _|
11     12      6+6       2     |_ _ _ _ _ _| |
12     28      23+5      2     |_ _ _ _ _ _ _|
...
For n = 6 the symmetric representation of sigma(6) has two subparts: [11, 1], so A000203(6) = 12 and A001227(6) = 2.
For n = 12 the symmetric representation of sigma(12) has two subparts: [23, 5], so A000203(12) = 28 and A001227(12) = 2. (End)
From _Hartmut F. W. Hoft_, Dec 26 2016: (Start)
Two examples of the general argument in the Comments section:
Rows 27 in A237048 and A249223 (4 parts)
i:  1  2 3 4 5 6 7 8 9 . . 12
27: 1  1 1 0 0 1                           1's in A237048 for odd divisors
    1 27 3     9                           odd divisors represented
27: 1  0 1 1 1 0 0 1 1 1 0 1               blocks forming parts in A249223
Rows 81 in A237048 and A249223 (5 parts)
i:  1  2 3 4 5 6 7 8 9 . . 12. . . 16. . . 20. . . 24
81: 1  1 1 0 0 1 0 0 1 0 0 0                          1's in A237048 f.o.d
    1 81 3    27     9                                odd div. represented
81: 1  0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 0 1  blocks fp in A249223
(End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Michel Marcus)

Row lengths of A237270 and of A379288.
Column 1 of A279387.
Partial sums give A237590.
Parity gives A347950.
Cf. A000203, A000265, A001065, A001227, A005279, A024916, A060831, A061345, A067742, A071561, A071562, A175254, A196020, A221529, A235791, A236104, A237048, A237591, A237593, A239657, A244050, A244971, A245092, A249223, A250068, A261699, A262045, A262612, A262626, A274824, A279387, A279693, A319073, A340583, A340846, A342344, A347186, A379288.
Cf. A027750, A174973 (2-dense numbers), A239663, A240062, A243982, A379379, A380580, A384149, A384222, A384225, A384226, A384230, A384930.

a237271[n_] := Length[a237270[n]] (* code defined in A237270 *)
Map[a237271, Range[90]] (* data *)
(* Hartmut F. W. Hoft, Jun 23 2014 *)
a[n_] := Module[{d = Partition[Divisors[n], 2, 1]}, 1 + Count[d, ?(OddQ[#[[2]]] && #[[2]] >= 2*#[[1]] &)]]; Array[a, 100] (* _Amiram Eldar,  Dec 22 2024 *)

fill(vcells, hga, hgb) = {ic = 1; for (i=1, #hgb, if (hga[i] < hgb[i], for (j=hga[i], hgb[i]-1, cell = vector(4); cell[1] = i - 1; cell[2] = j; vcells[ic] = cell; ic ++;););); vcells;}
findfree(vcells) = {for (i=1, #vcells, vcelli = vcells[i]; if ((vcelli[3] == 0) && (vcelli[4] == 0), return (i));); return (0);}
findxy(vcells, x, y) = {for (i=1, #vcells, vcelli = vcells[i]; if ((vcelli[1]==x) && (vcelli[2]==y) && (vcelli[3] == 0) && (vcelli[4] == 0), return (i));); return (0);}
findtodo(vcells, iz) = {for (i=1, #vcells, vcelli = vcells[i]; if ((vcelli[3] == iz) && (vcelli[4] == 0), return (i)); ); return (0);}
zcount(vcells) = {nbz = 0; for (i=1, #vcells, nbz = max(nbz, vcells[i][3]);); nbz;}
docell(vcells, ic, iz) = {x = vcells[ic][1]; y = vcells[ic][2]; if (icdo = findxy(vcells, x-1, y), vcells[icdo][3] = iz); if (icdo = findxy(vcells, x+1, y), vcells[icdo][3] = iz); if (icdo = findxy(vcells, x, y-1), vcells[icdo][3] = iz); if (icdo = findxy(vcells, x, y+1), vcells[icdo][3] = iz); vcells[ic][4] = 1; vcells;}
docells(vcells, ic, iz) = {vcells[ic][3] = iz; while (ic, vcells = docell(vcells, ic, iz); ic = findtodo(vcells, iz);); vcells;}
nbzb(n, hga, hgb) = {vcells = vector(sigma(n)); vcells = fill(vcells, hga, hgb); iz = 1; while (ic = findfree(vcells), vcells = docells(vcells, ic, iz); iz++;); zcount(vcells);}
lista(nn) = {hga = concat(heights(row237593(0), 0), 0); for (n=1, nn, hgb = heights(row237593(n), n); nbz = nbzb(n, hga, hgb); print1(nbz, ", "); hga = concat(hgb, 0););} \\ with heights() also defined in A237593; \\ Michel Marcus, Mar 28 2014

from sympy import divisors
def a(n: int) -> int:
    divs = list(divisors(n))
    d = [divs[i:i+2] for i in range(len(divs) - 1)]
    s = sum(1 for pair in d if len(pair) == 2 and pair[1] % 2 == 1 and pair[1] >= 2 * pair[0])
    return s + 1
print([a(n) for n in range(1, 80)])  # Peter Luschny, Aug 05 2025

a(n) = A001227(n) - A239657(n). - Omar E. Pol, Mar 23 2014
a(p^k) = k + 1, where p is an odd prime and k >= 0. - Hartmut F. W. Hoft, Dec 26 2016
Theorem: a(n) <= number of odd divisors of n (cf. A001227). The differences are in A239657. - N. J. A. Sloane, Jan 19 2021
a(n) = A340846(n) - A340833(n) + 1 (Euler's formula). - Omar E. Pol, Feb 01 2021
a(n) = A000005(n) - A243982(n). - Omar E. Pol, Aug 02 2025

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.

Original entry on oeis.org

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

Omar E. Pol, Jun 03 2025

The 2-dense sublists of divisors of n are the maximal sublists whose terms increase by a factor of at most 2.
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.
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.

			  ----------------------------------------------------------------
  |  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 = 14 the list of divisors of 14 is [1, 2, 7, 14]. There are two sublists of divisors of 14 whose terms increase by a factor of at most 2, they are [1, 2] and [7, 14]. Each sublist has two terms, so the row 14 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)

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

Row sums give A000005.

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

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

A384225 Irregular triangle read by rows: T(n,k) is the number of odd divisors in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4

Offset: 1

Omar E. Pol, Jun 16 2025

T(n,k) is the number of odd numbers in the k-th sublist of divisors of n whose terms increase by a factor of at most 2,
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.
At least for the first 1000 rows the row lengths give A237271.
Observation: at least the first 33 rows (or first 62 terms) coincide with A280940.

			  ------------------------------------------------------------------
  |  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, 1;        |  [1], [3];                   |     2     |
  |  4 |   1;           |  [1, 2, 4];                  |     1     |
  |  5 |   1, 1;        |  [1], [5];                   |     2     |
  |  6 |   2;           |  [1, 2, 3, 6];               |     1     |
  |  7 |   1, 1;        |  [1], [7];                   |     2     |
  |  8 |   1;           |  [1, 2, 4, 8];               |     1     |
  |  9 |   1, 1, 1;     |  [1], [3], [9];              |     3     |
  | 10 |   1, 1;        |  [1, 2], [5, 10];            |     2     |
  | 11 |   1, 1;        |  [1], [11];                  |     2     |
  | 12 |   2;           |  [1, 2, 3, 4, 6, 12];        |     1     |
  | 13 |   1, 1;        |  [1], [13];                  |     2     |
  | 14 |   1, 1;        |  [1, 2], [7, 14];            |     2     |
  | 15 |   1, 2, 1;     |  [1], [3, 5], [15];          |     3     |
  | 16 |   1;           |  [1, 2, 4, 8, 16];           |     1     |
  | 17 |   1, 1;        |  [1], [17];                  |     2     |
  | 18 |   3;           |  [1, 2, 3, 6, 9, 18];        |     1     |
  | 19 |   1, 1;        |  [1], [19];                  |     2     |
  | 20 |   2;           |  [1, 2, 4, 5, 10, 20];       |     1     |
  | 21 |   1, 1, 1, 1;  |  [1], [3], [7], [21];        |     4     |
   ...
For n = 14 the list of divisors of 14 is [1, 2, 7, 14]. There are two sublists of divisors of 14 whose terms increase by a factor of at most 2, they are [1, 2] and [7, 14]. Each sublist has only one odd number, so the row 14 is [1, 1].
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 odd numbers in the sublists are [1, 2, 1] respectively, so the row 15 is [1, 2, 1].
For n = 16 the list of divisors of 16 is [1, 2, 4, 8, 16]. There is only one sublist of divisors of 16 whose terms increase by a factor of at most 2, that is the same as the list of divisors of 16, which has five terms and only one odd number, so the row 16 is [1].

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

Row sums give A001227.

Cf. A000203, A027750, A174973 (2-dense numbers), A280940, A237271, A379288, A384149, A384222, A384226, A384928, A384930, A384931, A385000, A386984, A386989, A387030.

A384225row[n_] := Map[Count[#, _?OddQ] &, Split[Divisors[n], #2/# <= 2 &]];
Array[A384225row, 50] (* Paolo Xausa, Jul 08 2025 *)

A384226 Irregular triangle read by rows: T(n,k) is the sum of odd divisors in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 5, 4, 1, 7, 1, 1, 3, 9, 1, 5, 1, 11, 4, 1, 13, 1, 7, 1, 8, 15, 1, 1, 17, 13, 1, 19, 6, 1, 3, 7, 21, 1, 11, 1, 23, 4, 1, 5, 25, 1, 13, 1, 3, 9, 27, 8, 1, 29, 24, 1, 31, 1, 1, 3, 11, 33, 1, 17, 1, 12, 35, 13, 1, 37, 1, 19, 1, 3, 13, 39, 6, 1, 41, 32, 1, 43, 1, 11, 1, 32, 45, 1, 23, 1, 47, 4

Offset: 1

Omar E. Pol, Jun 24 2025

T(n,k) is the sum of odd numbers in the k-th sublist (or subsequence) of divisors of n such that the ratio of adjacent divisors in every sublist is at most 2.
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.
It shares the odd-indexed rows with A384149.
At least for the first 1000 rows the row lengths give A237271.

			  --------------------------------------------------------------------
  |  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,  3;         |  [1], [3];                   |     2     |
  |  4 |   1;             |  [1, 2, 4];                  |     1     |
  |  5 |   1,  5;         |  [1], [5];                   |     2     |
  |  6 |   4;             |  [1, 2, 3, 6];               |     1     |
  |  7 |   1,  7;         |  [1], [7];                   |     2     |
  |  8 |   1;             |  [1, 2, 4, 8];               |     1     |
  |  9 |   1,  3,  9;     |  [1], [3], [9];              |     3     |
  | 10 |   1,  5;         |  [1, 2], [5, 10];            |     2     |
  | 11 |   1, 11;         |  [1], [11];                  |     2     |
  | 12 |   4;             |  [1, 2, 3, 4, 6, 12];        |     1     |
  | 13 |   1, 13;         |  [1], [13];                  |     2     |
  | 14 |   1,  7;         |  [1, 2], [7, 14];            |     2     |
  | 15 |   1,  8, 15;     |  [1], [3, 5], [15];          |     3     |
  | 16 |   1;             |  [1, 2, 4, 8, 16];           |     1     |
  | 17 |   1, 17;         |  [1], [17];                  |     2     |
  | 18 |  13;             |  [1, 2, 3, 6, 9, 18];        |     1     |
  | 19 |   1, 19;         |  [1], [19];                  |     2     |
  | 20 |   6;             |  [1, 2, 4, 5, 10, 20];       |     1     |
  | 21 |   1,  3,  7, 21; |  [1], [3], [7], [21];        |     4     |
   ...
For n = 14 the list of divisors of 14 is [1, 2, 7, 14]. There are two sublists of divisors of 14 whose terms increase by a factor of at most 2, they are [1, 2] and [7, 14]. The sums of odd terms in the sublists are [1], [7] respectively, so the row 14 is [1, 7].
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 sums of terms in the sublists are [1, 8, 15] respectively, so the row 15 is [1, 8, 15].
For n = 16 the list of divisors of 16 is [1, 2, 4, 8, 16]. There is only one sublist of divisors of 16 whose terms increase by a factor of at most 2, that is the same as the list of divisors of 16, so the row 16 is [1].
For n = 2350 the list of divisors of 2350 is [1, 2, 5, 10, 25, 47, 50, 94, 235, 470, 1175, 2350]. There are five sublists of divisors of 2350 whose terms increase by a factor of at most 2, they are [1, 2], [5, 10], [25, 47, 50, 94], [235, 470], [1175, 2350]. The sums of odd terms in the sublists are [1, 5, 72, 235, 1175] respectively, so the row 2350 is [1, 5, 72, 235, 1175].

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

Row sums give A000593.

Cf. A174973 (2-dense numbers), A237271, A237593, A379288, A384149, A384222, A384225, A384928.

A384226row[n_] := Map[Total[Select[#, OddQ]] &, Split[Divisors[n], #2/# <= 2 &]];
Array[A384226row, 50] (* Paolo Xausa, Jul 08 2025 *)

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.

Original entry on oeis.org

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

Omar E. Pol, Jul 19 2025

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.

			  ---------------------------------------------------------------------
  |  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.

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

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.

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

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).

A384928 Number of 2-dense sublists of divisors of the n-th triangular number.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 4, 1, 1, 3, 4, 1, 2, 3, 4, 1, 2, 5, 6, 3, 1, 5, 4, 1, 1, 5, 6, 1, 4, 5, 6, 1, 1, 5, 6, 1, 2, 3, 6, 1, 2, 7, 8, 3, 1, 3, 4, 1, 1, 5, 6, 3, 4, 7, 3, 1, 1, 5, 4, 1, 2, 3, 8, 1, 1, 7, 8, 3, 3, 5, 6, 1, 2, 3, 6, 1, 4, 5, 8, 1, 1, 7, 4, 1, 1, 7, 6, 1, 4, 5, 3, 3, 3, 5, 8, 1, 2, 5, 5, 1, 6

Offset: 0

Omar E. Pol, Aug 08 2025

nonn

By definition a(n) is also the number of 2-dense sublists of divisors of the n-th generalized hexagonal number.
In a sublist of divisors of k the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of k.
The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.
Conjecture: all odd indexed terms are odd.

			For n = 5 the 5th triangular number is 15. The list of divisors of 15 is [1, 3, 5, 15]. There are three 2-dense sublists of divisors of 15, they are [1], [3, 5], [15], so a(5) = 3.
For n = 12 the 12th triangular number is 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], so a(12) = 2. Note that 78 is also the first practical number A005153 not in the sequence of the 2-dense numbers A174973.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000217, A005153, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384930, A384931, A386984 (a bisection), A386989.

A384928[n_] := Length[Split[Divisors[PolygonalNumber[n]], #2 <= 2*# &]];
Array[A384928, 100, 0] (* Paolo Xausa, Aug 14 2025 *)

a(n) = A237271(A000217(n)) for n >= 1 (conjectured).

A386984 Number of 2-dense sublists of divisors of the n-th hexagonal number.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 3, 1, 5, 3, 5, 1, 5, 1, 5, 1, 5, 1, 3, 1, 7, 3, 3, 1, 5, 3, 7, 1, 5, 1, 3, 1, 7, 3, 5, 1, 3, 1, 5, 1, 7, 1, 7, 1, 5, 3, 5, 1, 5, 1, 7, 3, 5, 1, 7, 1, 7, 5, 7, 1, 5, 3, 3, 1, 7, 1, 7, 1, 7, 5, 5, 1, 7, 1, 7, 3, 5, 1, 3, 1, 9, 3, 7, 1, 7, 1, 7, 1, 5, 1, 5, 1, 9, 3, 3, 1, 3, 1, 9, 1

Offset: 0

Omar E. Pol, Aug 11 2025

In a sublist of divisors of k the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of k.
The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.
Conjecture: all terms are odd.

			For n = 3 the third positive hexagonal number is 15. The list of divisors of 15 is [1, 3, 5, 15]. There are three 2-dense sublists of divisors of 15, they are [1], [3, 5], [15], so a(3) = 3.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Bisection of A384928.

Cf. A000384, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384930, A384931, A386989.

A386984[n_] := Length[Split[Divisors[PolygonalNumber[6, n]], #2 <= 2*# &]];
Array[A386984, 100, 0] (* Paolo Xausa, Aug 29 2025 *)

a(n) = A237271(A000384(n)) for n >= 1 (conjectured).

A379379 Sum of odd divisors of n except the "e" odd divisors described in A005279.

Original entry on oeis.org

1, 1, 4, 1, 6, 1, 8, 1, 13, 6, 12, 1, 14, 8, 19, 1, 18, 1, 20, 1, 32, 12, 24, 1, 31, 14, 40, 1, 30, 1, 32, 1, 48, 18, 41, 1, 38, 20, 56, 1, 42, 1, 44, 12, 49, 24, 48, 1, 57, 31, 72, 14, 54, 1, 72, 1, 80, 30, 60, 1, 62, 32, 95, 1, 84, 1, 68, 18, 96, 41, 72, 1, 74

Offset: 1

Omar E. Pol, Dec 21 2024

Shares infinitely many terms with A000593.
a(n) = A000593(n) if n is not in A005279.
a(n) < A000593(n) if n is in A005279.
Conjectures from Omar E. Pol, Aug 27 2025: (Start)
a(n) is the sum of the smallest numbers of the 2-dense sublists of divisors of n.
a(n) is the sum of the divisors p of n such that p is greater than twice the adjacent previous divisor of n. (End)

Row sums of A379288.

Cf. A001227, A000593, A005279, A182469, A237270, A237271, A237593, A239657, A379374, A379384, A379461.

a[n_] := Module[{d = Partition[Divisors[n], 2, 1]}, 1 + Total[Select[d, OddQ[#[[2]]] && #[[2]] >= 2*#[[1]] &][[;; , 2]]]]; Array[a, 100] (* Amiram Eldar, Dec 22 2024 *)

More terms from Alois P. Heinz, Dec 22 2024

A387030 Irregular triangle read by rows: T(n,k) is the number of primes in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 0, 1, 1, 0, 1, 0, 0, 2, 0, 1, 3, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 2, 0, 2, 0, 1, 1, 1, 0, 1, 1, 0, 2, 0, 1, 3, 0, 1, 1, 1, 0, 2, 0, 1, 1, 0, 1, 2, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1

Offset: 1

Omar E. Pol, Aug 13 2025

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.
It is conjectured that row lengths are given by A237271.

			Triangle begins:
  0;
  1;
  0, 1;
  1;
  0, 1;
  2;
  0, 1;
  1;
  0, 1, 0;
  1, 1;
  0, 1;
  2;
  0, 1;
  1, 1;
  0, 2, 0;
  ...
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]. There is a prime number in each sublist, so row 10 is [1, 1].
For n = 15 the list of divisors of 15 is [1, 3, 5, 15]. There are three 2-dense sublists of divisors of 15, they are [1], [3, 5], [15]. Only the second sublist contains primes, so row 15 is [0, 2, 0].

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

Row sums give A001221.

Cf. A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384930.

A387030row[n_] := Map[Count[#, _?PrimeQ] &, Split[Divisors[n], #2 <= 2*# &]];
Array[A387030row, 50] (* Paolo Xausa, Aug 19 2025 *)

A379374 Irregular triangle read by rows in which row n lists the divisors of n except the divisors "e" described in A005279.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 11, 1, 2, 12, 1, 13, 1, 2, 7, 14, 1, 3, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 6, 18, 1, 19, 1, 2, 4, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 24, 1, 5, 25, 1, 2, 13, 26, 1, 3, 9, 27

Offset: 1

Omar E. Pol, Dec 21 2024

Observation: the sequence of the number of odd terms in row n coincides with at least the first 10000 terms of A237271.

The observation is true for all numbers. For a proof see A379288. - Hartmut F. W. Hoft, Jan 25 2025

			Triangle begins:
  1;
  1,  2;
  1,  3;
  1,  2,  4;
  1,  5;
  1,  2,  6;
  1,  7;
  1,  2,  4,  8;
  1,  3,  9;
  1,  2,  5, 10;
  1, 11;
  1,  2, 12;
  1, 13;
  1,  2,  7, 14;
  1,  3, 15;
  1,  2,  4,  8, 16;
  1, 17;
  1,  2,  6, 18;
  1, 19;
  1,  2,  4, 10, 20;
  ...

Subsequence of A027750.
Row sums give A379384.
Odd terms give A379288.
Cf. A001227, A005279, A010814, A237271, A237593, A239657, A237271, A379379.

row[n_] := Module[{d = Partition[Divisors[n], 2, 1]}, Join[{1}, Select[d, #[[2]] >= 2*#[[1]] &][[;; , 2]]]]; Table[row[n], {n, 1, 27}] // Flatten (* Amiram Eldar, Dec 22 2024 *)

More terms from Alois P. Heinz, Dec 21 2024

cp's OEIS Frontend

A237271 Number of parts in the symmetric representation of sigma(n).

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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.

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A384225 Irregular triangle read by rows: T(n,k) is the number of odd divisors in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

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A384226 Irregular triangle read by rows: T(n,k) is the sum of odd divisors in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

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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.

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A384928 Number of 2-dense sublists of divisors of the n-th triangular number.

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A386984 Number of 2-dense sublists of divisors of the n-th hexagonal number.

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A379379 Sum of odd divisors of n except the "e" odd divisors described in A005279.

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