A174973 -id:A174973 - cp's OEIS frontend

A384149 Irregular triangle T(n, k) in which row n gives the 2-densely-aggregated composition of sigma(n).

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

Offset: 1

Peter Munn, May 22 2025

We form the 2-densely-aggregated composition of sigma(n) = A000203(n) by listing the divisors of n in increasing order and assigning adjacent divisors for summation in the same aggregate if (and only if) they differ by a factor of less than or equal to 2. The ordering of the aggregate sums in the composition follows the ordering of the summed divisors.
We follow here the use of 2-dense/2-densely reported in the comments of A174973.
If n is in A174973 then row n has length 1 and its sole term is sigma(n).
From empirical evidence of the first 1000 rows, reinforced by some known related properties, it looks readily credible that if we take the average of row n and its reversal we get A237270 row n, a palindromic composition of sigma(n) that was defined using Dyck paths. Row lengths are therefore conjectured to be in the database as A237271.

			For row 9: the ordered divisors of 9 are (1, 3, 9). Adjacent divisors differ by a factor of 3, which is greater than 2, so each divisor is trivially summed into a separate aggregate and the 2-densely-aggregated composition of sigma(9) is (1, 3, 9).
For row 12, the ordered divisors of 12 are (1, 2, 3, 4, 6, 12). Every pair of adjacent divisors differs by a factor <= 2, so they are summed in a single aggregate and the 2-densely-aggregated composition of sigma(12) is (1+2+3+4+6+12) = (28).
For row 10, the ordered divisors of 10 are (1, 2, 5, 10). The adjacent divisors (1, 2) and (5, 10) differ by a factor of 2, but (2, 5) differ by a larger factor, so there are 2 aggregates and the 2-densely-aggregated composition of sigma(10) is (1+2, 5+10) = (3, 15).
For 29029 = 7 * 11 * 13 * 29, the 2-densely-aggregated composition of sigma(29029) is (1, 7+11+13, 29, 77+91+143+203+319+377, 1001, 2233+2639+4147, 29029) = (1, 31, 29, 1210, 1001, 9019, 29029). Note that this composition is not in ascending order.
Triangle begins:
  row
   1  1,
   2  3,
   3  1, 3,
   4  7,
   5  1, 5,
   6  12,
   7  1, 7,
   8  15,
   9  1, 3, 9,
  10  3, 15,
  11  1, 11,
  12  28,
  13  1, 13,
  14  3, 21,
  15  1, 8, 15,
  16  31,
  ...
If we take the average of row 9, (1, 3, 9) and its reversal, (9, 3, 1), we get (5, 3, 5), which is A237270 row 9. Doing the same for row 10, (3, 15), we get (9, 9), which is A237270 row 10.

Cf. A000203, A027750, A174973, A237270, A237271.

t384149[n_] := Module[{dL = Divisors[n]}, Map[#[[1]] &, Map[Apply[Plus, #] &, Split[Transpose[{dL, Append[Rest[dL], 2 n + 1]}], #[[2]] <= 2 #[[1]] &]]]] (* row n of triangle *)
a384149[n_] := Flatten[Map[t384149, Range[n]]]
a384149[45] (* Hartmut F. W. Hoft, Jun 07 2025 *)

A379288 Irregular triangle read by rows in which row n lists the odd divisors of n excluding odd divisors e for which there exists another divisor j with j < e < 2*j.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 5, 1, 1, 7, 1, 1, 3, 9, 1, 5, 1, 11, 1, 1, 13, 1, 7, 1, 3, 15, 1, 1, 17, 1, 1, 19, 1, 1, 3, 7, 21, 1, 11, 1, 23, 1, 1, 5, 25, 1, 13, 1, 3, 9, 27, 1, 1, 29, 1, 1, 31, 1, 1, 3, 11, 33, 1, 17, 1, 5, 35, 1, 1, 37, 1, 19, 1, 3, 13, 39, 1, 1, 41, 1, 1, 43

Offset: 1

Omar E. Pol, Dec 21 2024

The excluded divisors are the odd divisors e listed in A005279.
Conjecture 1: the row lengths are given by A237271 (true for at least the first 10000 terms of A237271)
From Hartmut F. W. Hoft, Jan 09 2025: (Start)
Proof of Conjecture 1:
An entire part of SRS(n), n = 2^k * q with k >= 0 and q odd, up to the diagonal is described in row n of A249223 by a 1 in position d, an odd divisor of n, 0's in positions d-1 and 2^(k+1) * f, f >= d an odd divisor of n, and nonzero numbers that increase or decrease by 1 in between.
The odd divisors e of n with d < e < 2^(k+1) * f are the "e" odd divisors of A005279 since for divisor s of n, d < s < e < 2*s < 2^(k+1) * f holds.
The odd divisors u of n greater than A003056(n) are encoded by the 2^(k+1) * f above as u = q/f and odd divisors d < A003056(n) are also encoded as 2^(k+1) * q/d. Then odd divisors e of n with q/f < e < 2^(k+1) * q/d are the "e" odd divisors of A005279 since for divisor t of n,  q/f < t < e < 2*t < 2^(k+1) * q/d holds.
For a part containing the diagonal the inequalities above hold on the respective sides of the diagonal.
As a consequence the number of entries in row n of this triangle equals A237271(n). (End)
From Omar E. Pol, Jun 26 2025: (Start)
Conjecture 2: T(n,m) is the smallest number in the m-th 2-dense sublist 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.
If the conjecture is true so row sums give A379379 and the row lengths give A237271, and the same row lengths have the sequences A384222, A384225 and A384226. Also the conjecture of A384149 should be true.
Observation: at least for the first 5000 rows (the first 15542 terms) the conjecture 2  coincides with the definition from the Name section and the row lengths give A237271.
An example of the conjecture 2, for n = 1..24 is as shown below:
  -------------------------------------------------------------------
  |  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 |   1;             |  [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 |   1;             |  [1, 2, 3, 4, 6, 12];        |     1     |
  | 13 |   1, 13;         |  [1], [13];                  |     2     |
  | 14 |   1,  7;         |  [1, 2], [7, 14];            |     2     |
  | 15 |   1,  3, 15;     |  [1], [3, 5], [15];          |     3     |
  | 16 |   1;             |  [1, 2, 4, 8, 16];           |     1     |
  | 17 |   1, 17;         |  [1], [17];                  |     2     |
  | 18 |   1;             |  [1, 2, 3, 6, 9, 18];        |     1     |
  | 19 |   1, 19;         |  [1], [19];                  |     2     |
  | 20 |   1;             |  [1, 2, 4, 5, 10, 20];       |     1     |
  | 21 |   1,  3,  7, 21; |  [1], [3], [7], [21];        |     4     |
  | 22 |   1, 11;         |  [1, 2], [11, 22];           |     2     |
  | 23 |   1, 23;         |  [1], [23];                  |     2     |
  | 24 |   1;             |  [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 2-dense sublists of divisors of 10, they are [1, 2], [5, 10]. The smallest numbers in the sublists are [1, 5] respectively, so the row 10 is [1, 5].
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]. The smallest numbers in the sublists are [1, 3, 15] respectively, so the row 15 is [1, 3, 15].
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 2-dense sublists of divisors of 78, they are [1, 2, 3, 6] and [13, 26, 39, 78]. The smallest numbers in the sublists are [1, 13] respectively, so the row 78 is [1, 13].
(End)
Conjecture 3: T(n,m) is the m-th divisor p of n such that p is greater than twice the adjacent previous divisor of n. - Omar E. Pol, Aug 02 2025

These are the odd terms of A379374.
Subsequence of A182469.
Row sums give A379379.
Cf. A001227, A005279, A237270, A237271, A237593, A239657, A379384, A379461, A379634.
Cf. A000203, A003056, A249223.
Cf. A005153, A174973, A384149, A384222, A384225, A384226, A385000.

row[n_] := Module[{d = Partition[Divisors[n], 2, 1]}, Select[Join[{1}, Select[d, #[[2]] >= 2*#[[1]] &][[;; , 2]]], OddQ]]; Table[row[n], {n, 1, 50}] // Flatten (* Amiram Eldar, Dec 22 2024 *)

More terms from Amiram Eldar, Dec 22 2024

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 *)

A280107 Numbers m with the property that the symmetric representation of sigma(m) has four parts.

Original entry on oeis.org

21, 27, 33, 39, 51, 55, 57, 65, 69, 75, 85, 87, 93, 95, 105, 111, 115, 119, 123, 125, 129, 133, 141, 145, 155, 159, 161, 175, 177, 183, 185, 201, 203, 205, 213, 215, 217, 219, 230, 235, 237, 245, 249, 250, 253, 259, 265, 267, 287, 290, 291, 295, 301, 303, 305, 309, 310, 319, 321, 327, 329

Offset: 1

Omar E. Pol, Dec 27 2016

nonn

From Hartmut F. W. Hoft, Jan 27 2018: (Start)
Let n = 2^k * t where k >= 0 and t is odd, and let D be the set of divisors of t less than r(n) = floor((sqrt(8*n+1) - 1)/2). The following statements are equivalent:
(1)  There is exactly one d in D such that 2^(k+1) * d < e where e in D is the next odd divisor larger than d, and the largest divisor f in D satisfies 2^(k+1) * f <= r(n).
(2)  The symmetric representation of sigma(n) consists of four parts.
The property says that the first part of the symmetric representation of n consists of the first 2^(k+1) * d - 1 legs and that the second part starts with leg e and ends with leg 2^(k+1) * f - 1 before or at the middle of the Dyck path (see A237048 and A249223) on the diagonal. Together with their symmetric pair they form the four parts. (End)

			a(1) = 21 because it is the smallest number n whose symmetric representation of sigma(n) has four parts. Note that the sum of the parts is 11 + 5 + 5 + 11 = 32, equaling the sum of the divisors of 21: aigma(21) = 1 + 3 + 7 + 21 = 32.
From _Hartmut F. W. Hoft_, Jan 27 2018: (Start)
230 = 2*5*23 is the first even number since 2^2 < 5, 2^2 * 5 < 23, and row 230 in A237048 has 20 entries with 1's in positions 1, 4, 5, and 20.
Prime number 3 can be a factor for an even number in this sequence as 12246=2*3*13*157 demonstrates with the four parts 12252, 1020, 1020, and 12252 in the symmetric representation of sigma(12246) defined by 1's in positions 1, 3, 4, 12, 13, 39, 52, 156 in row 12246 of A237048; each of the four parts has maximum width 2 and the two central parts meet on the diagonal at 8492. (End)

Column 4 of A240062.
First differs from A264102 at a(10).
Numbers n such that the symmetric representation of sigma(n) has k parts (k = 1..4): A174973 = A238443, A239929, A279102, this sequence.
Cf. A237270, A237271 (number of parts), A237591, A237593, A239665, A245092, A262626.
Cf. A237048, A249223. - Hartmut F. W. Hoft, Jan 27 2018

(* Function a237270[] is defined in A237270 *)
a280107[m_, n_] := Select[Range[m, n], Length[a237270[#]]==4&]
a280107[1, 329] (* data *)
(* Implementation of the property in the Comment section *)
evenPart[n_] := Module[{f=First[FactorInteger[n]]}, If[First[f]!=2, 1, First[f]^Last[f]]]
fourPartsQ[n_] := Module[{e=evenPart[n], oddPart, r=(Sqrt[8*n + 1] - 1)/2, dL}, oddPart=n/evenPart[n]; dL=Select[Divisors[oddPart], #1, 2*e*Last[dL]<=r && Length[Select[2*e*Most[dL]-Rest[dL], #<0&]]==1, False]];
Select[Range[329], fourPartsQ] (* data *)
(* Hartmut F. W. Hoft, Jan 27 2018 *)

A082662 Numbers k such that the odd part of k is less than sqrt(2k).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 20, 24, 28, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 144, 160, 176, 192, 208, 224, 240, 256, 272, 288, 304, 320, 336, 352, 368, 384, 400, 416, 432, 448, 464, 480, 496, 512, 544, 576, 608, 640, 672, 704, 736, 768, 800

Offset: 1

Naohiro Nomoto, May 18 2003

Theorem: The following eight definitions are equivalent.
(P1) Numbers k such that the odd part of k (A000265(k)) is < sqrt(2k).
  (P1) is the new definition, repeated here for convenience. Note that this is not the same as saying A000265(k) < A172471(k), since A172471(k) = floor(sqrt(2*k)).
(P2) Numbers k such that the odd divisors of k are < sqrt(2k).
  (P2) and (P1) are obviously equivalent.
(P3) The numbers 1, S_0, S_1, S_2, ..., where
  S_m = { 2^(m+1)*(2^m+i) : i = 0 .. 3*2^m - 1 }.
So S_0 = {2,4,6}, S_1 = {8,12,16,20,24,28}, S_2 = {32,40,48,...,120}, S_3 = {128,144,...,496}, ...
  The proof that (P3) and (P1) are the same sequence is not difficult and will be added later. (P3) is equivalent to a formula stated without proof (it may have been only an empirical observation) in the original version of this entry.
(P4) Numbers k such that the odd part of k is <= A003056(k).
  That is, the odd part of k is <= floor((sqrt(1+8*n)-1)/2). It is more difficult to show this is equivalent to (P1), but it is true.
(P5) Numbers k such that the odd divisors of k are <= A003056(k).
  (P5) and (P4) are obviously equivalent.
(P6) Numbers k such that A001227(k) = A082647(k).
  (P6) was the original definition. In words, it says that the number of odd divisors of k is equal to the number of ways to write k as a sum of an odd number of consecutive positive integers, or equivalently as a sum of d consecutive positive integers for some d dividing k.  To show that (P6) is equivalent to (P1) one makes use of the Hirschhorn-Hirschhorn article.
(P7) Numbers k such that the odd part of k is <= the sum of divisors of the even part.
  (P7) was contributed by Jaycob Coleman, Jun 21 2014. To show (P7) is equivalent to (P1), write k as 2^m*s where s is odd.  Equality holds if and only if k is an even perfect number.
(P8) Numbers k such that A000265(k) <= A000203(A006519(k)) or also such that A000265(k) <= A038712(k).
  (P8) was contributed by Michel Marcus, Aug 14 2014. It is a restatement of (P7).
(End of theorem)
A further equivalent property, (P9), follows at once from (P4). This was conjectured by  Omar E. Pol, Apr 18 2017
(P9) These are the numbers k such that the sequence of successive widths in the symmetric representation of sigma(k) is unimodal.
Yet another equivalent property:
(P10) Numbers k >= 1 such if k = i + (i+1) + (i+2) + ... + (i+j-1) for some i >= 1 and j >= 1 then j is odd [Caballero, 2019]. - Michel Marcus, Jan 16 2020
This is a subsequence of A005153. - Jaycob Coleman, Jun 21 2014
The complement of this sequence is A281005. - Omar E. Pol, Apr 18 2017
Subsequence of A174973. - Omar E. Pol, Feb 01 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000
José Manuel Rodríguez Caballero, Integers Which Cannot Be Partitioned Into an Even Number of Consecutive Parts, INTEGERS, Volume 19 (2019), #A20.
M. D. Hirschhorn and P. M. Hirschhorn, Partitions into Consecutive Parts, Mathematics Magazine: 2003, Volume 76, Number 4, pp. 306-308.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 38.
Eric Weisstein's World of Mathematics, Even Part
Eric Weisstein's World of Mathematics, Odd Part

Cf. A281005 (complement).

Cf. A000196, A000203, A000265, A001227, A003056, A005153, A006519, A038712, A053644, A082647, A116882, A172471, A174973, A237593, A279387.

cnt[n_] := DivisorSum[n, Boole[OddQ[#] && #>Sqrt[2n]]&]; Select[Range[800], cnt[#]==0&] (* Jean-François Alcover, Feb 16 2017 *)

isok(n) = my(q = sqrt(2*n)); (sumdiv(n, d, (d%2) && (d < q)) == sumdiv(n, d, d%2)); \\ Michel Marcus, Jul 04 2014

G.f. = 1 + (1/(1-x)^2) * Sum_{m >= 0} (2^(m+1)*x^(3*2^m-2) * ( x^(3*2^m)*(2^(m+2)*(x-1)-x) - 2^m*(x-1) + x ) ). (This follows from (P3).) :w
- N. J. A. Sloane, Feb 02 2021
a(n+1) = a(n) + A053644(A000196(2*a(n))). - Peter Munn, Oct 03 2023

Edited by N. J. A. Sloane, Jan 28 2021: Replaced original indirect definition by simple direct definition; rearranged comments; provided proofs (not yet included here) that the various definitions are equivalent

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).

A047836 "Nullwertzahlen" (or "inverse prime numbers"): n=p1p2p3p4p5...pk, where pi are primes with p1 <= p2 <= p3 <= p4 ...; then p1 = 2 and p1p2...*pi >= p(i+1) for all i < k.

Original entry on oeis.org

2, 4, 8, 12, 16, 24, 32, 36, 40, 48, 56, 60, 64, 72, 80, 84, 96, 108, 112, 120, 128, 132, 144, 160, 168, 176, 180, 192, 200, 208, 216, 224, 240, 252, 256, 264, 280, 288, 300, 312, 320, 324, 336, 352, 360, 384, 392, 396, 400, 408, 416, 420, 432, 440, 448

Offset: 1

Thomas Kantke (bytes.more(AT)ibm.net)

Start with n and reach 2 by repeatedly either dividing by d where d <= the square root or by adding or subtracting 1. The division steps are free, but adding or subtracting 1 costs 1 point. The "value" of n (A047988) is the smallest cost to reach 2. Sequence gives numbers with value 0.
a(n) is also the length of the largest Dyck path of the symmetric representation of sigma of the n-th number whose symmetric representation of sigma has only one part. For an illustration see A317305. (Cf. A237593.) - Omar E. Pol, Aug 25 2018
This sequence can be defined equivalently as the increasing terms of the set containing 2 and all the integers such that if n is in the set, then all m * n are in the set for all m <= n. - Giuseppe Melfi, Oct 21 2019
The subsequence giving the largest term with k prime factors (k >= 1) starts 2, 4, 12, 132, 17292, 298995972, ... . - Peter Munn, Jun 04 2020

			Starting at 24 we divide by 3, 2, then 2, reaching 2.

T. D. Noe, Table of n, a(n) for n = 1..1000
Thomas Kantke, Das Spiel Minimum und die Zerlegung natürlicher Zahlen, Spektrum der Wissenschaft, No. 4, 1993, pp. 11-13.
Andreas Weingartner, Uniform distribution of alpha*n modulo one for a family of integer sequences, arXiv:2303.16819 [math.NT], 2023.

Cf. A047984, A047985, A047986, A047987, A047988, A052287, A237593, A317305.

import Data.List.Ordered (union)
a047836 n = a047836_list !! (n-1)
a047836_list = f [2] where
   f (x:xs) = x : f (xs `union` map (x *) [2..x])
-- Reinhard Zumkeller, Jun 25 2015, Sep 28 2011

nMax = 100; A174973 = Select[Range[10*nMax], AllTrue[Rest[dd = Divisors[#]] / Most[dd], Function[r, r <= 2]]&]; a[n_] := 2*A174973[[n]]; Array[a, nMax] (* Jean-François Alcover, Nov 10 2016, after Reinhard Zumkeller *)

a(n) = 2 * A174973(n). - Reinhard Zumkeller, Sep 28 2011

The number of terms <= x is c*x/log(x) + O(x/(log(x))^2), where c = 0.612415..., and a(n) = C*n*log(n*log(n)) + O(n), where C = 1/c = 1.63287... This follows from the formula just above. - Andreas Weingartner, Jun 30 2021

More terms from David W. Wilson

A238524 Numbers n such that the symmetric representation of sigma(n) is formed by two or more parts.

Original entry on oeis.org

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92

Offset: 1

Omar E. Pol, Mar 06 2014

nonn

Complement of A174973.
First differs from A237046 at a(48).
First differs from A237287 at a(55).
For more information see A237270.
From Hartmut F. W. Hoft, Nov 27 2014: (Start)
Suppose n = 2^m * p1^e1 *...* pk^ek where p1 < ... < pk are the odd prime factors of n, m>=0 and all ej>0. Equivalent to the property of numbers in this sequence are:
(a) The number of 1's in odd positions equals the number of 1's in even positions in the n-th row of triangle A237048 through an index of the form 2^(m+1)*q where q is an odd divisor of n.
(b) There is one odd prime factor pj of n satisfying  pj > 2^(m+1) * product_{iAlso numbers n for which the n-th row in irregular triangle A249223 contains a zero.
(End)

			9 is in the sequence because the symmetric representation of sigma(9) = 13 is formed by three parts: [5, 3, 5], as shown below in the first quadrant:
.        5
.    _ _ _ _ _
.   |_ _ _ _ _|
.             |_ _ 3
.             |_  |       Sigma(9) = 5 + 3 + 5 = 13
.               |_|_ _
.                   | |
.                   | |
.                   | | 5
.                   | |
.                   |_|
.
From _Hartmut F. W. Hoft_, Nov 27 2014: (Start)
Number 78 = 2 * 3 * 13 has 1's in the 78th row of triangle A237048 at indices 1, 3, 4, 12 where 12 = 2^2*3 < 13. The symmetric representation of sigma(78) has two regions that meet at a point on the diagonal (width 0) and their third leg has width 2. Note also that 78 is the smallest number in this sequence for which width 0 occurs at an index that is not a power of 2.
(End)

Cf. A174973, A196020, A236104, A235791, A237046, A237287, A237591, A237593, A237270, A237271.

(* sequence of numbers k for m <= k <= n having two or more parts *)
(* Function a237270[] is defined in A237270 *)
a238524[m_, n_]:=Select[Range[m, n], Length[a237270[#]]>=2&]
a238524[1, 260] (* data *)
(* Hartmut F. W. Hoft, Jul 07 2014 *)
(* function for the alternate description of the sequence *)
(* functions row[ ] & a237048[ ] are defined in A237048 *)
zero249223Q[n_] := Module[{i=2, bound=row[n], width=1}, While[width>=1 && i<=bound, width += (-1)^(i+1) * a237048[n, i]; i++]; width==0]
Select[Range[1, 100], zero249223Q] (* data *)
(* Hartmut F. W. Hoft, Nov 27 2014 *)

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cp's OEIS Frontend

A384149 Irregular triangle T(n, k) in which row n gives the 2-densely-aggregated composition of sigma(n).

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A379288 Irregular triangle read by rows in which row n lists the odd divisors of n excluding odd divisors e for which there exists another divisor j with j < e < 2*j.

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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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A280107 Numbers m with the property that the symmetric representation of sigma(m) has four parts.

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A082662 Numbers k such that the odd part of k is less than sqrt(2k).

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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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A047836 "Nullwertzahlen" (or "inverse prime numbers"): n=p1p2p3p4p5...pk, where pi are primes with p1 <= p2 <= p3 <= p4 ...; then p1 = 2 and p1p2...*pi >= p(i+1) for all i < k.