A384930 -id:A384930 - 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 *)

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).

A384931 Number of 2-dense sublists of divisors of the number of partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 2, 1, 1, 1, 3, 2, 3, 1, 5, 6, 4, 4, 5, 1, 2, 4, 3, 4, 1, 5, 4, 7, 2, 4, 9, 10, 4, 9, 2, 6, 9, 3, 1, 9, 4, 11, 8, 4, 3, 3, 8, 12, 4, 11, 7, 10, 5, 3, 7, 2, 2, 1, 8, 5, 6, 8, 5, 2, 1, 3, 10, 6, 1, 6, 8, 7, 1, 1, 4, 2, 7, 9, 3, 4, 9, 6, 2

Offset: 0

Omar E. Pol, Jul 30 2025

In a 2-dense 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.

			For n = 7 the number of partitions of 7 is A000041(7) = 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(7) = 3.
For n = 19 the number of partitions of 19 is A000041(19) = 490. The list of divisors of 490 is [1, 2, 5, 7, 10, 14, 35, 49, 70, 98, 245, 490]. There are four 2-dense sublists of divisors of 490, they are [1, 2], [5, 7, 10, 14], [35, 49, 70, 98], [245, 490], so a(19) = 4.

Alois P. Heinz, Table of n, a(n) for n = 0..4000

Cf. A000041, A174973 (2-dense numbers), A237271, A384149, A384222, A384225, A384226, A384930.

A384931[n_] := Length[Split[Divisors[PartitionsP[n]], #2 <= 2*# &]];
Array[A384931, 100, 0] (* Paolo Xausa, Aug 28 2025 *)

a(n) = A237271(A000041(n)). Conjectured.

More terms from Alois P. Heinz, Jul 30 2025

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).

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 *)

A386989 Irregular triangle read by rows: T(n,k) is the product of terms in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 1, 3, 8, 1, 5, 36, 1, 7, 64, 1, 3, 9, 2, 50, 1, 11, 1728, 1, 13, 2, 98, 1, 15, 15, 1024, 1, 17, 5832, 1, 19, 8000, 1, 3, 7, 21, 2, 242, 1, 23, 331776, 1, 5, 25, 2, 338, 1, 3, 9, 27, 21952, 1, 29, 810000, 1, 31, 32768, 1, 3, 11, 33, 2, 578, 1, 35, 35, 10077696, 1, 37, 2, 722, 1, 3, 13, 39, 2560000

Offset: 1

Omar E. Pol, Aug 12 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:
   1;
   2;
   1,  3;
   8;
   1,  5;
  36;
   1,  7;
  64;
   1,  3,  9;
   2, 50;
  ...
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]. The product of terms are 1*2 = 2 and 5*10 = 50 respectively, so the row 10 of the triangle is [2, 50].

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

Row products give A007955.

Cf. A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384928, A384930, A384931, A386984.

A386989row[n_] :=Times @@@ Split[Divisors[n], #2/# <= 2 &];
Array[A386989row, 50] (* Paolo Xausa, Aug 29 2025 *)

A386992 Irregular triangle read by rows: T(n,k) is the number of nonprimes in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 0, 2, 1, 0, 3, 1, 0, 1, 1, 1, 1, 0, 4, 1, 0, 1, 1, 1, 0, 1, 4, 1, 0, 4, 1, 0, 4, 1, 0, 0, 1, 1, 1, 1, 0, 6, 1, 0, 1, 1, 1, 1, 0, 1, 1, 4, 1, 0, 5, 1, 0, 5, 1, 0, 0, 1, 1, 1, 1, 0, 1, 7, 1, 0, 1, 1, 1, 0, 0, 1, 6, 1, 0, 5, 1, 0, 2, 2, 1, 2, 1, 1, 1, 1, 0, 8, 1, 0, 1, 1, 1, 2, 1, 0, 0, 1, 2, 2, 1, 0

Offset: 1

Omar E. Pol, Aug 23 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:
  1;
  1;
  1, 0;
  2;
  1, 0;
  2;
  1, 0;
  3;
  1, 0, 1;
  1, 1;
  1, 0;
  4;
  1, 0;
  1, 1;
  1, 0, 1;
  ...
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 nonprime 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 first and the third sublists contain nonprimes, so row 15 is [1, 0, 1].

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

Row sums give A033273.

Cf. A018252, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384928, A384930, A384931, A386984, A386993, A387030.

A386992row[n_] := Map[Count[#, _?(!PrimeQ[#] &)] &, Split[Divisors[n], #2 <= 2*# &]];
Array[A386992row, 50] (* Paolo Xausa, Aug 28 2025 *)

T(n,k) = A384222(n,k) - A387030(n,k).

A386993 Number of 2-dense sublists of divisors of the n-th squarefree number.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 23 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.

			For n = 11 the 11th squarefree 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(11) = 3.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A005117, A174973 (2-dense numbers), A237271, A379288, A380328, A384149, A384222, A384225, A384226, A384928, A384930, A384931, A386984, A386992, A387030.

Map[Length[Split[Divisors[#], #2 <= 2*# &]] &, Select[Range[150], SquareFreeQ]] (* Paolo Xausa, Aug 29 2025 *)

a(n) = A237271(A005117(n)). (conjectured).

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

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A384931 Number of 2-dense sublists of divisors of the number of partitions of n.

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

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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.

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A386989 Irregular triangle read by rows: T(n,k) is the product of terms in the k-th 2-dense sublist of divisors of n, with n >= 1, k >= 1.

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