A239934 -id:A239934 - cp's OEIS frontend

A244894 Composite numbers n with the property that the symmetric representation of sigma(n) has two parts.

10, 14, 22, 26, 34, 38, 44, 46, 52, 58, 62, 68, 74, 76, 78, 82, 86, 92, 94, 102, 106, 114, 116, 118, 122, 124, 134, 136, 138, 142, 146, 148, 152, 158, 164, 166, 172, 174, 178, 184, 186, 188, 194, 202, 206, 212, 214, 218, 222, 226, 232, 236, 244, 246, 248, 254, 258, 262, 268, 274, 278, 282, 284, 292, 296, 298, 302, 314, 316, 318, 326, 328, 332, 334, 344, 346, 348, 354, 356, 358

Offset: 1

Omar E. Pol, Jul 07 2014

nonn

Even numbers in A239929.

By definition the two parts of the symmetric representation of sigma(n) are sigma(n)/2 and sigma(n)/2.

			Illustration of the symmetric representation of sigma(n) in the second quadrant for the first four elements of this sequence: [10, 14, 22, 26].
.
.                             _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                            |  _ _ _ _ _ _ _ _ _ _ _ _ _|
.                            | |
.                            | |
.                            | |  _ _ _ _ _ _ _ _ _ _ _ _
.                      21 _ _| | |  _ _ _ _ _ _ _ _ _ _ _|
.                        |_ _ _| | |
.                     _ _|       | |
.                   _|     18 _ _| |
.                  |         |_ _ _|
.            21 _ _|        _|
.              | |        _|
.     _ _ _ _ _| | 18 _ _|                _ _ _ _ _ _ _ _
.    |  _ _ _ _ _|   | |                 |  _ _ _ _ _ _ _|
.    | |      _ _ _ _| |                 | |
.    | |     |  _ _ _ _|             12 _| |
.    | |     | |                       |_ _|  _ _ _ _ _ _
.    | |     | |                 12 _ _|     |  _ _ _ _ _|
.    | |     | |              _ _ _| |    9 _| |
.    | |     | |             |  _ _ _|  9 _|_ _|
.    | |     | |             | |      _ _| |
.    | |     | |             | |     |  _ _|
.    | |     | |             | |     | |
.    | |     | |             | |     | |
.    | |     | |             | |     | |
.    | |     | |             | |     | |
.    |_|     |_|             |_|     |_|
.
n:    26      22              14      10
.
Sigma(10) =  9 +  9 = 18.
Sigma(14) = 12 + 12 = 24.
Sigma(22) = 18 + 18 = 36.
Sigma(26) = 21 + 21 = 42.
.

R. J. Mathar, Table of n, a(n) for n = 1..999

Cf. A237271 (number of parts), A237270, A237593, A238443, A238524, A239660, A239929, A239932, A239934, A245092, A262626, A280107 (4 parts).

Extended by R. J. Mathar, Oct 04 2018

A244970 Total number of regions after n-th stage in the diagram of the symmetric representation of sigma on the four quadrants.

Original entry on oeis.org

1, 2, 6, 7, 11, 12, 16, 17, 25, 29, 33, 34, 38, 42, 50, 51, 55, 56, 60, 61, 73, 77, 81, 82, 90, 94, 106, 107, 111, 112, 116, 117, 129, 133, 141, 142, 146, 150, 162, 163, 167, 168, 172, 176, 184, 188, 192, 193, 201, 209, 221, 225, 229, 230, 242, 243, 255, 259, 263, 264

Offset: 1

Omar E. Pol, Jul 08 2014

nonn

Partial sums of A244971.
If we use toothpicks of length 1/2, so the area of the central square is equal to 1. The total area of the structure after n-th stage is equal to A024916(n), the sum of all divisors of all positive integers <= n, hence the total area of the n-th set of symmetric regions added at n-th stage is equal to sigma(n) = A000203(n), the sum of divisors of n.
If we use toothpicks of length 1, so the number of cells (and the area) of the central square is equal to 4. The number of cells (and the total area) of the structure after n-th stage is equal to 4*A024916(n) = A243980(n), hence the number of cells (and the total area) of the n-th set of symmetric regions added at n-th stage is equal to 4*A000203(n) = A239050(n).
a(n) is also the total number of terraces of the stepped pyramid with n levels described in A244050. - Omar E. Pol, Apr 20 2016

			Illustration of the structure after 15 stages (contains 50 regions):
.
.                   _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.                  |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.               _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _
.             _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_
.           _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_
.          |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  |
.     _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _
.    | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | |
.    | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | |
.    | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | |
.    | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | |
.    | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | |
.    | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | |
.    | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | |
.    | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | |
.    | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | |
.    | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | |
.    | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | |
.    | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | |
.    | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | |
.    | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | |
.    | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | |
.    |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_|
.          | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |
.          |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _|
.            |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|
.              |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|
.                  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
The diagram is also the top view of the stepped pyramid with 15 levels described in A244050. - _Omar E. Pol_, Apr 20 2016

Cf. A000203, A004125, A024916, A196020, A235791, A236104, A237270, A237271, A237590, A237591, A237593, A239050, A239660, A239931-A239934, A243980, A244050, A244360-A244363, A244370, A244371, A244971, A245092.

A340846 a(n) is the number of edges in the diagram of the symmetric representation of sigma(n).

Original entry on oeis.org

4, 6, 8, 10, 10, 12, 12, 14, 16, 16, 14, 18, 14, 18, 22, 22, 16, 22, 16, 22, 26, 22, 18, 26, 24, 22, 28, 28, 20, 30, 20, 30, 30, 24, 28, 32, 22, 26, 32, 34, 22, 34, 22, 34, 38, 28, 24, 38, 32, 40, 34, 36, 24, 38, 38, 42, 36, 30, 26, 42, 26, 30, 46, 42, 40, 44, 28

Offset: 1

Omar E. Pol, Jan 24 2021

nonn

Since the diagram is symmetric so all terms are even numbers.

For another version with subparts see A340848 from which first differs at a(6).

			Illustration of initial terms:
.                                                          _ _ _ _
.                                            _ _ _        |_ _ _  |_
.                                _ _ _      |_ _ _|             |   |_
.                      _ _      |_ _  |_          |_ _          |_ _  |
.              _ _    |_ _|_        |_  |           | |             | |
.        _    |_  |       | |         | |           | |             | |
.       |_|     |_|       |_|         |_|           |_|             |_|
.
n:       1      2        3          4           5               6
a(n):    4      6        8         10          10              12
.
For n = 6 the diagram has 12 edges so a(6) = 12.
On the other hand the diagram has 12 vertices and only one part or region, so applying Euler's formula we have that a(6) = 12 + 1 - 1 = 12.
.                                                  _ _ _ _ _
.                            _ _ _ _ _            |_ _ _ _ _|
.        _ _ _ _            |_ _ _ _  |                     |_ _
.       |_ _ _ _|                   | |_                    |_  |
.               |_                  |_  |_ _                  |_|_ _
.                 |_ _                |_ _  |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   |_|                   |_|                     |_|
.
n:              7                    8                      9
a(n):          12                   14                     16
.
For n = 9 the diagram has 16 edges so a(9) = 16.
On the other hand the diagram has 14 vertices and three parts or regions, so applying Euler's formula we have that a(9) = 14 + 3 - 1 = 16.
Another way for the illustration of initial terms is as follows:
--------------------------------------------------------------------------
.  n  a(n)                             Diagram
--------------------------------------------------------------------------
            _
   1   4   |_|  _
              _| |  _
   2   6     |_ _| | |  _
                _ _|_| | |  _
   3   8       |_ _|  _| | | |  _
                  _ _|  _| | | | |  _
   4  10         |_ _ _|  _|_| | | | |  _
                    _ _ _|  _ _| | | | | |  _
   5  10           |_ _ _| |    _| | | | | | |  _
                      _ _ _|  _|  _|_| | | | | | |  _
   6  12             |_ _ _ _|  _|  _ _| | | | | | | |  _
                        _ _ _ _|  _|  _ _| | | | | | | | |  _
   7  12               |_ _ _ _| |  _|  _ _|_| | | | | | | | |  _
                          _ _ _ _| |  _| |  _ _| | | | | | | | | |  _
   8  14                 |_ _ _ _ _| |_ _| |  _ _| | | | | | | | | | |  _
                            _ _ _ _ _|  _ _|_|  _ _|_| | | | | | | | | | |
   9  16                   |_ _ _ _ _| |  _|  _|  _ _ _| | | | | | | | | |
                              _ _ _ _ _| |  _|  _|    _ _| | | | | | | | |
  10  16                     |_ _ _ _ _ _| |  _|     |  _ _|_| | | | | | |
                                _ _ _ _ _ _| |      _| |  _ _ _| | | | | |
  11  14                       |_ _ _ _ _ _| |  _ _|  _| |  _ _ _| | | | |
                                  _ _ _ _ _ _| |  _ _|  _|_|  _ _ _|_| | |
  12  18                         |_ _ _ _ _ _ _| |  _ _|  _ _| |  _ _ _| |
                                    _ _ _ _ _ _ _| |  _| |    _| |  _ _ _|
  13  14                           |_ _ _ _ _ _ _| | |  _|  _|  _| |
                                      _ _ _ _ _ _ _| | |_ _|  _|  _|
  14  18                             |_ _ _ _ _ _ _ _| |  _ _|  _|
                                        _ _ _ _ _ _ _ _| |  _ _|
  15  22                               |_ _ _ _ _ _ _ _| | |
                                          _ _ _ _ _ _ _ _| |
  16  22                                 |_ _ _ _ _ _ _ _ _|
...

Cf. A237271 (number of parts or regions).
Cf. A340833 (number of vertices).
Cf. A340848 (number of edges in the diagram with subparts).
Cf. A317109 (total number of edges in the unified diagram).
Cf. A239931-A239934 (illustration of first 32 diagrams).
Cf. A000203, A005843, A196020, A236104, A235791, A237048, A237270, A237590, A237591, A237593, A239660, A245092, A262626, A340847.

a(n) = A340833(n) + A237271(n) - 1 (Euler's formula).

More terms from Omar E. Pol, Oct 28 2021

A243982 Number of divisors of n minus the number of parts in the symmetric representation of sigma(n).

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 0, 3, 0, 2, 0, 5, 0, 2, 1, 4, 0, 5, 0, 5, 0, 2, 0, 7, 0, 2, 0, 5, 0, 7, 0, 5, 0, 2, 1, 8, 0, 2, 0, 7, 0, 7, 0, 4, 3, 2, 0, 9, 0, 3, 0, 4, 0, 7, 0, 7, 0, 2, 0, 11, 0, 2, 1, 6, 0, 7, 0, 4, 0, 5, 0, 11, 0, 2, 2, 4, 1, 6, 0, 9, 0, 2, 0, 11, 0, 2, 0, 7, 0, 11, 1, 4, 0, 2, 0, 11, 0, 3, 1, 8, 0, 6, 0, 7

Offset: 1

Omar E. Pol, Jun 16 2014

nonn

Conjecture: a(n) is the number of divisors r of n such that r is not greater than twice the adjacent previous divisor of n. - Omar E. Pol, Aug 04 2025

			For n = 9 the divisors of 9 are [1, 3, 9] and the parts of the symmetric representation of sigma(9) are [5, 3, 5]. In both cases there are three elements, so a(9) = 3 - 3 = 0.
For n = 10 the four divisors of 10 are [1, 2, 5, 10] and the two parts of the symmetric representation of sigma(10) are [9, 9], so a(10) = 4 - 2 = 2.

Antti Karttunen, Table of n, a(n) for n = 1..5000 (computed from the b-file of A237271 provided by Michel Marcus)

Cf. A000005, A000203, A071561, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239657, A239660, A239931-A239934.

(* Function a237270[] is defined in A237270 *)
a243982[n_]:=Length[Divisors[n] - Length[a237270[n]]
a243982[m_, n_]:=Map[a243982, Range[m,n]]
a243982[1, 104]] (* data *)
(* Hartmut F. W. Hoft, Sep 19 2014 *)

a(n) = A000005(n) - A237271(n).

a(94)-a(95) corrected by Omar E. Pol, Jul 02 2014

A317306 Powers of 2 and even perfect numbers.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 28, 32, 64, 128, 256, 496, 512, 1024, 2048, 4096, 8128, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33550336, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589869056, 8589934592

Offset: 1

Omar E. Pol, Aug 23 2018

Numbers k such that the symmetric representation of sigma(k) has only one part, and apart from the central width, the rest of the widths are 1's.

Note that the above definition implies that the central width of the symmetric representation of sigma(k) is 1 or 2. For powers of 2 the central width is 1. For even perfect numbers the central width is 2 (see example).

			Illustration of initial terms:
.        _ _   _   _   _               _                       _       _
.    1  |_| | | | | | | |             | |                     | |     | |
.    2  |_ _|_| | | | | |             | |                     | |     | |
.        _ _|  _|_| | | |             | |                     | |     | |
.    4  |_ _ _|    _|_| |             | |                     | |     | |
.        _ _ _|  _|  _ _|             | |                     | |     | |
.    6  |_ _ _ _|  _|                 | |                     | |     | |
.        _ _ _ _| |                   | |                     | |     | |
.    8  |_ _ _ _ _|              _ _ _| |                     | |     | |
.                               |  _ _ _|                     | |     | |
.                              _| |                           | |     | |
.                            _|  _|                           | |     | |
.                        _ _|  _|                             | |     | |
.                       |  _ _|                               | |     | |
.                       | |                          _ _ _ _ _| |     | |
.        _ _ _ _ _ _ _ _| |                         |  _ _ _ _ _|     | |
.   16  |_ _ _ _ _ _ _ _ _|                         | |    _ _ _ _ _ _| |
.                                                _ _| |   |  _ _ _ _ _ _|
.                                            _ _|  _ _|   | |
.                                           |    _|    _ _| |
.                                          _|  _|     |  _ _|
.                                         |  _|      _| |
.                                    _ _ _| |      _|  _|
.                                   |  _ _ _|  _ _|  _|
.                                   | |       |  _ _|
.                                   | |  _ _ _| |
.                                   | | |  _ _ _|
.        _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | |
.   28  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.                                       | |
.                                       | |
.        _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.   32  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
The diagram shows the first eight terms of the sequence. The symmetric representation of sigma has only one part, and apart from the central width, the rest of the widths are 1's.
A317307(n) is the area (or the number of cells) in the n-th region of the diagram.

Union of A000079 and A000396 assuming there are no odd perfect numbers.
Subsequence of A174973.
Cf. A249351 (the widths).
Cf. A317307(n) = sigma(a(n)).
Cf. A000203, A000225, A139256, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A262626.

A342344 Number of parts in the symmetric representation of antisigma(n).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Mar 08 2021

nonn

In order to construct this sequence and the diagram of the symmetric representation of antisigma(n) = A024816(n) we use the following rules:
At stage 1 in the first quadrant of the square grid we draw the symmetric representation of sigma(n) using the two Dyck paths described in the rows n and n-1 of A237593. The area of the region that is below the symmetric representation of sigma(n) equals A024916(n-1).
At stage 2 we draw a pair of orthogonal line segments (if it's necessary) such that in the drawing appears totally formed a square n X n. The area of the region that is above the symmetric representation of sigma(n) equals A004125(n). Then we draw a zig-zag path with line segments of length 1 from (0,n-1) to (n-1,0) such that appears a staircase with n-1 steps. The area of the region (or regions) that is below the symmetric representation of sigma(n) and above the staircase equals A244048(n) = A153485(n-1). The area of the region that is below the staircase equals A000217(n-1).
At stage 3 we turn OFF the cells of the symmetric representation of sigma(n) and also the cells that are below the staircase. Then we turn ON the rest of the cells that are in the square n X n. The result is that the ON cell form the diagram of the symmetric representation of antisigma(n) = A024816(n). See the Example section.
For n >= 7; if A237271(n) = 1 or n is a term of A262259 then a(n) = 2 otherwise a(n) = 1.

			Illustration of the symmetric representation of antisigma(n) = AS(n) = A024816(n), for n = 1..6:
.                                                             y|        _ _
.                                              y|      _ _     |  _ _  |_  |
.                                 y|      _     |  _ _|   |    | |_  |   |_|
.                      y|    _     |  _  |_|    | |_     _|    |   |_|_ _
.             y|        |  _|_|    | |_|_       |   |_  |      |     |_  |
.      y|      |        | |_|      |   |_|      |     |_|      |       |_|
.       |_ _   |_ _ _   |_ _ _ _   |_ _ _ _ _   |_ _ _ _ _ _   |_ _ _ _ _ _ _
.          x        x          x            x              x                x
.
n:        1       2         3           4             5               6
a(n):     0       0         2           3             1               3
AS(n):    0       0         2           3             9               9
.
Illustration of the symmetric representation of antisigma(n) = AS(n) = A024816(n), for n = 7..9:
.                                                y|          _ _ _ _
.                          y|          _ _ _      |  _ _ _ _|       |
.      y|        _ _ _      |  _ _ _  |     |     | |_       _ _    |
.       |  _ _ _|     |     | |_    | |_    |     |   |_    |_  |   |
.       | |_          |     |   |_  |_  |_ _|     |     |_    |_|  _|
.       |   |_       _|     |     |_  |_ _        |       |_      |
.       |     |_    |       |       |_    |       |         |_    |
.       |       |_  |       |         |_  |       |           |_  |
.       |         |_|       |           |_|       |             |_|
.       |_ _ _ _ _ _ _ _    |_ _ _ _ _ _ _ _ _    |_ _ _ _ _ _ _ _ _ _
.                      x                     x                       x
.
n:              7                    8                      9
a(n):           1                    2                      1
AS(n):         20                   21                     32
.
For n = 9 the figures 1, 2 and 3 below show respectively the three stages described in the Comments section as follows:
.
.   y|_ _ _ _ _ 5            y|_ _ _ _ _ _ _ _ _      y|          _ _ _ _
.    |_ _ _ _ _|              |_ _ _ _ _|       |      |  _ _ _ _|       |
.    |         |_ _ 3         | |_      |_ _ R  |      | |_       _ _    |
.    |         |_  |          |   |_    |_  |   |      |   |_    |_  |   |
.    |           |_|_ _ 5     |     |_ T  |_|_ _|      |     |_    |_|  _|
.    |               | |      |       |_      | |      |       |_      |
.    |      Q        | |      |         |_    | |      |         |_    |
.    |               | |      |    W      |_  | |      |           |_  |
.    |               | |      |             |_| |      |             |_|
.    |_ _ _ _ _ _ _ _|_|_     |_ _ _ _ _ _ _ _|_|_     |_ _ _ _ _ _ _ _ _ _
.                       x                        x                        x
.         Figure 1.                Figure 2.                Figure 3.
.         Symmetric                Symmetric                Symmetric
.       representation           representation           representation
.         of sigma(9)              of sigma(9)            of antisigma(9)
.       A000203(9) = 13          A000203(9) = 13          A024816(9) = 32
.           and of                   and of
.     Q = A024916(8) = 56      R = A004125(9) = 12
.                              T = A244048(9) = 20
.                              T = A153485(8) = 20
.                              W = A000217(8) = 36
.
Note that the symmetric representation of antisigma(9) contains a hole formed by three cells because these three cells were the central part of the symmetric representation of sigma(9).

Cf. A000203, A000217, A000290, A004125, A024816, A024916, A153485, A174973, A236104, A237270, A237271, A237593, A238443, A239660, A239931, A239932, A239933, A239934, A244048, A262259.

A244250 Triangle read by rows in which row n lists the widths in the first octant of the symmetric representation of sigma(n).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 26 2014

For the definition of k-th width of the symmetric representation of sigma(n) see A249351.
Row n list the first n terms of the n-th row of A249351.
It appears that the leading diagonal is also A067742 (which was conjectured by Michel Marcus in the entry A237593 and checked with two Mathematica functions up to n = 100000 by Hartmut F. W. Hoft).
For more information see A237591, A237593.

			Triangle begins:
1;
1, 1;
1, 1, 0;
1, 1, 1, 1;
1, 1, 1, 0, 0;
1, 1, 1, 1, 1, 2;
1, 1, 1, 1, 0, 0, 0;
1, 1, 1, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 0, 0, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 0;
1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0;
1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2;
1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0;
...

Cf. A000203, A067742, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A238443, A239660, A239932-A239934, A240542, A241008, A241010, A244360, A244361, A245685, A246955, A246956, A247687, A249223, A249351, A250068, A250070, A250071.

A266094 a(n) is the sum of the divisors of the smallest number k such that the symmetric representation of sigma(k) has n parts.

Original entry on oeis.org

1, 4, 13, 32, 104, 228, 576, 1408, 4104, 9824, 19152, 39816, 82944, 196992, 441294, 881280, 1911168, 4539024

Offset: 1

Omar E. Pol, Dec 21 2015

For more information see A239663 and A239665.

			Illustration of the symmetric representation of sigma(9):
.
.     _ _ _ _ _ 5
.    |_ _ _ _ _|
.              |_ _ 3
.              |_  |
.                |_|_ _ 5
.                    | |
.                    | |
.                    | |
.                    | |
.                    |_|
.
For n = 3 we have that 9 is the smallest number whose symmetric representation of sigma has three parts: [5, 3, 5], so a(3) = 5 + 3 + 5 = 13, equaling the sum of divisors of 9: sigma(9) = 1 + 3 + 9 = 13.
For n = 7 we have that 357 is the smallest number whose symmetric representation of sigma has seven parts: [179, 61, 29, 38, 29, 61, 179], so a(7) = 179 + 61 + 29 + 38 + 29 + 61 + 179 = 576, equaling the sum of divisors of 357: sigma(357) = 1 + 3 + 7 + 17 + 21 + 51 + 119 + 357 = 576.

Cf. A000203, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A239931-A239934, A239663, A239665, A240062, A245092, A262626.

a(n) = A000203(A239663(n)).

a(14)-a(18) from Omar E. Pol, Jul 21 2018

A299778 Irregular triangle read by rows: T(n,k) is the part that is adjacent to the k-th peak of the largest Dyck path of the symmetric representation of sigma(n), or T(n,k) = 0 if the mentioned part is already associated to a previous peak or if there is no part adjacent to the k-th peak, with n >= 1, k >= 1.

Original entry on oeis.org

1, 3, 2, 2, 7, 0, 3, 3, 12, 0, 0, 4, 0, 4, 15, 0, 0, 5, 3, 5, 9, 0, 9, 0, 6, 0, 0, 6, 28, 0, 0, 0, 7, 0, 0, 7, 12, 0, 12, 0, 8, 8, 0, 0, 8, 31, 0, 0, 0, 0, 9, 0, 0, 0, 9, 39, 0, 0, 0, 0, 10, 0, 0, 0, 10, 42, 0, 0, 0, 0, 11, 5, 0, 5, 0, 11, 18, 0, 0, 0, 18, 0, 12, 0, 0, 0, 0, 12, 60, 0, 0, 0, 0, 0, 13, 0, 5, 0, 0, 13

Offset: 1

Omar E. Pol, Apr 03 2018

For the definition of "part" of the symmetric representation of sigma see A237270.

For more information about the mentioned Dyck paths see A237593.

			Triangle begins (rows 1..28):
   1;
   3;
   2,  2;
   7,  0;
   3,  3;
  12,  0,  0;
   4,  0,  4;
  15,  0,  0;
   5,  3,  5;
   9,  0,  9,  0;
   6,  0,  0,  6;
  28,  0,  0,  0;
   7,  0,  0,  7;
  12,  0, 12,  0;
   8,  8,  0,  0,  8;
  31,  0,  0,  0,  0;
   9,  0,  0,  0,  9;
  39,  0,  0,  0,  0;
  10,  0,  0,  0, 10;
  42,  0,  0,  0,  0;
  11,  5,  0,  5,  0, 11;
  18,  0,  0,  0, 18,  0;
  12,  0,  0,  0,  0, 12;
  60,  0,  0,  0,  0,  0;
  13,  0,  5,  0,  0, 13;
  21,  0,  0,  0  21,  0;
  14,  6,  0,  6,  0, 14;
  56,  0,  0,  0,  0,  0,  0;
  ...
Illustration of first 50 terms (rows 1..16 of triangle) in an irregular spiral which can be find in the top view of the pyramid described in A244050:
.
.               12 _ _ _ _ _ _ _ _
.                 |  _ _ _ _ _ _ _|_ _ _ _ _ _ _ 7
.                 | |             |_ _ _ _ _ _ _|
.              0 _| |                           |
.               |_ _|9 _ _ _ _ _ _              |_ _ 0
.         12 _ _|     |  _ _ _ _ _|_ _ _ _ _ 5      |_ 0
.    0 _ _ _| |    0 _| |         |_ _ _ _ _|         |
.     |  _ _ _|  9 _|_ _|                   |_ _ 3    |_ _ _ 7
.     | |    0 _ _| |   12 _ _ _ _          |_  |         | |
.     | |     |  _ _|  0 _|  _ _ _|_ _ _ 3    |_|_ _ 5    | |
.     | |     | |    0 _|   |     |_ _ _|         | |     | |
.     | |     | |     |  _ _|           |_ _ 3    | |     | |
.     | |     | |     | |    3 _ _        | |     | |     | |
.     | |     | |     | |     |  _|_ 1    | |     | |     | |
.    _|_|    _|_|    _|_|    _|_| |_|    _|_|    _|_|    _|_|    _
.   | |     | |     | |     | |         | |     | |     | |     | |
.   | |     | |     | |     |_|_ _     _| |     | |     | |     | |
.   | |     | |     | |    2  |_ _|_ _|  _|     | |     | |     | |
.   | |     | |     |_|_     2    |_ _ _|  0 _ _| |     | |     | |
.   | |     | |    4    |_               7 _|  _ _|0    | |     | |
.   | |     |_|_ _     0  |_ _ _ _        |  _|    _ _ _| |     | |
.   | |    6      |_      |_ _ _ _|_ _ _ _| |  0 _|    _ _|0    | |
.   |_|_ _ _     0  |_   4        |_ _ _ _ _|  _|     |    _ _ _| |
.  8      | |_ _   0  |                     15|      _|   |  _ _ _|
.         |_    |     |_ _ _ _ _ _            |  _ _|  0 _| |      0
.        8  |_  |_    |_ _ _ _ _ _|_ _ _ _ _ _| |    0 _|  _|
.          0  |_ _|  6            |_ _ _ _ _ _ _|  _ _|  _|  0
.            0    |                             28|  _ _|  0
.                 |_ _ _ _ _ _ _ _                | |    0
.                 |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _| |
.                8                |_ _ _ _ _ _ _ _ _|
.                                                    31
.
The diagram contains A237590(16) = 27 parts.
For the construction of the spiral see A239660.

Index entries for sequences related to sigma(n)

Row sums give A000203.
Row n has length A003056(n).
Column k starts in row A000217(k).
Nonzero terms give A237270.
The number of nonzero terms in row n is A237271(n).
Column 1 is A241838.
The triangle with n rows contain A237590(n) nonzero terms.
Cf. A296508 (analog for subparts).
Cf. A024916, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A239657, A239660, A239931-A239934, A240542, A244050, A245092, A250068, A250070, A261699, A262626, A279387, A279388, A279391, A280850, A280851.

A317307 Sum of divisors of powers of 2 and sum of divisors of even perfect numbers.

Original entry on oeis.org

1, 3, 7, 12, 15, 31, 56, 63, 127, 255, 511, 992, 1023, 2047, 4095, 8191, 16256, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67100672, 67108863, 134217727, 268435455, 536870911, 1073741823, 2147483647, 4294967295, 8589934591, 17179738112, 17179869183

Offset: 1

Omar E. Pol, Aug 25 2018

Sum of divisors of the numbers k such that the symmetric representation of sigma(k) has only one part, and apart from the central width, the rest of the widths are 1's.

Note that the above definition implies that the central width of the symmetric representation of sigma(k) is 1 or 2. For powers of 2 the central width is 1. For even perfect numbers the central width is 2 (see example).

			Illustration of initial terms. a(n) is the area (or the number of cells) of the n-th region of the diagram:
.        _ _   _   _   _               _                       _       _
.   1   |_| | | | | | | |             | |                     | |     | |
.   3   |_ _|_| | | | | |             | |                     | |     | |
.        _ _|  _|_| | | |             | |                     | |     | |
.   7   |_ _ _|    _|_| |             | |                     | |     | |
.        _ _ _|  _|  _ _|             | |                     | |     | |
.  12   |_ _ _ _|  _|                 | |                     | |     | |
.        _ _ _ _| |                   | |                     | |     | |
.  15   |_ _ _ _ _|              _ _ _| |                     | |     | |
.                               |  _ _ _|                     | |     | |
.                              _| |                           | |     | |
.                            _|  _|                           | |     | |
.                        _ _|  _|                             | |     | |
.                       |  _ _|                               | |     | |
.                       | |                          _ _ _ _ _| |     | |
.        _ _ _ _ _ _ _ _| |                         |  _ _ _ _ _|     | |
.  31   |_ _ _ _ _ _ _ _ _|                         | |    _ _ _ _ _ _| |
.                                                _ _| |   |  _ _ _ _ _ _|
.                                            _ _|  _ _|   | |
.                                           |    _|    _ _| |
.                                          _|  _|     |  _ _|
.                                         |  _|      _| |
.                                    _ _ _| |      _|  _|
.                                   |  _ _ _|  _ _|  _|
.                                   | |       |  _ _|
.                                   | |  _ _ _| |
.                                   | | |  _ _ _|
.        _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | |
.   56  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.                                       | |
.                                       | |
.        _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.   63  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
The diagram shows the first eight terms of the sequence. The symmetric representation of sigma of the numbers A317306: 1, 2, 4, 6, 8, 16, 28, 32, ..., has only one part, and apart from the central width, the rest of the widths are 1's.

Union of nonzero terms of A000225 and A139256.
Odd terms give the nonzeros terms of A000225.
Even terms give A139256.
Subsequence of A317305.
Cf. A249351 (the widths).
Cf. A000203, A000396, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A262626, A317306.

DivisorSigma[1, #] &@ Union[2^Range[0, Floor@ Log2@ Last@ #], #] &@ Array[2^(# - 1) (2^# - 1) &@ MersennePrimeExponent@ # &, 7] (* Michael De Vlieger, Aug 25 2018, after Robert G. Wilson v at A000396 *)

a(n) = A000203(A317306(n)).

cp's OEIS Frontend

A244894 Composite numbers n with the property that the symmetric representation of sigma(n) has two parts.

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A244970 Total number of regions after n-th stage in the diagram of the symmetric representation of sigma on the four quadrants.

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A340846 a(n) is the number of edges in the diagram of the symmetric representation of sigma(n).

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A243982 Number of divisors of n minus the number of parts in the symmetric representation of sigma(n).

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A317306 Powers of 2 and even perfect numbers.

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A342344 Number of parts in the symmetric representation of antisigma(n).

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A244250 Triangle read by rows in which row n lists the widths in the first octant of the symmetric representation of sigma(n).

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A266094 a(n) is the sum of the divisors of the smallest number k such that the symmetric representation of sigma(k) has n parts.

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A317307 Sum of divisors of powers of 2 and sum of divisors of even perfect numbers.

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