A238443 -id:A238443 - cp's OEIS frontend

A244579 Numbers k with the property that the number of parts in the symmetric representation of sigma(k) equals the number of divisors of k.

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 79, 81, 83, 85, 87, 89, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131, 133, 137, 139, 141, 145

Offset: 1

Omar E. Pol, Jul 02 2014

nonn

Numbers n such that A243982(n) = 0.
First differs from A151991 at a(25).
Let n = 2^m * q with m >= 0 and q odd. Let c_n denote the count of regions in the symmetric representation of sigma(n), which is determined by the positions of 1's in the n-th row of A237048. The maximum of c_n occurs when odd and even positions of 1's alternate implying that all regions have width 1, denoted by w_n = 1. When m > 0 then sigma_0(n) > sigma_0(q) and c_n = sigma_0(n) is impossible. Therefore, exactly those odd n with w_n = 1 are in this sequence. Furthermore, since the 1's in A237048 represent the odd divisors of n, their odd-even alternation expresses the property 2*f < g for any two adjacent divisors f < g of odd number n; in other words, this sequence is also the complement of A090196 relative to the odd numbers. This last property permits computations of elements in this sequence faster than with function a244579, which is based on Dyck paths. - Hartmut F. W. Hoft, Oct 11 2015
From Hartmut F. W. Hoft, Dec 06 2016: (Start)
Also, integers n such that for any pair a < b of divisors of n the inequality 2*a < b holds (hence n is odd).
Let 1 = d_1 < ... < d_k = n be all (odd) divisors of n. The property 2*d_i < d_(i+1), for 1 <= i < k, is equivalent for the 1's in the n-th row of A249223 to be in positions 1 = d_1 < 2 < d_2 < 2*d_2 < ... < d_i <2*d_i < d_(i+1) < ... where 2*d_i represents the odd divisor e_i with d_i * e_i = n. In other words, the odd divisors are the number of parts in the symmetric representation of sigma(n). The rightmost 1 in the n-th row occurs in an odd (even) position when k is odd (even).
As a consequence this sequence is also the complement of A090196 in the set of odd numbers. (End)

			9 is in the sequence because the parts of the symmetric representation of sigma(9) are [5, 3, 5] and the divisors of 9 are [1, 3, 9] and in both cases there is the same number of elements: A237271(9) = A000005(9) = 3.
See the link for a diagram of the symmetric representations of sigma for sequence data listed above. The symmetric representations of sigma(a(35)) = sigma(81) = sigma(3^4) consists of 5 regions whose areas are [41, 15, 9, 15, 41] and computed as 41 = (3^4+3^0)/2, 15 = (3^3+3^1)/2, and 9 = 3^2 for the central area. Observe also that the 81st row in triangle A237048 is [ 1 1 1 0 0 1 0 0 1 0 0 0 ] with the 1's in positions 1, 2, 3, 6, and 9. This is the largest count for the symmetric regions of sigma shown in the diagram. - _Hartmut F. W. Hoft_, Oct 11 2015

Hartmut F. W. Hoft, Illustration of the symmetric representations of sigma for sequence data

Cf. A000005, A001227, A005279, A071561, A071562, A090196, A000203, A196020, A236104, A237048, A237270, A237271, A237593, A238443, A238524, A239657, A239929, A239663, A240062, A241558, A241559, A243982, A245092.

(* Function a237270[] is defined in A237270 *)
a244579[m_, n_] := Select[Range[m,n], Length[a237270[#]] == Length[Divisors[#]]&]
a244579[1, 150] (* data *)
(* Hartmut F. W. Hoft, Sep 19 2014 *)
(* alternative function using the divisor property *)
divisorPairsQ[n_] := Module[{d=Divisors[n]}, Select[2*Most[d] - Rest[d], # >= 0&] == {}]
a244579Alt[m_?OddQ, n_] := Select[Range[m, n, 2], divisorPairsQ]
a244579Alt[1, 145] (* data *)
(* Hartmut F. W. Hoft, Oct 11 2015 *)

A237271(a(k)) = A000005(a(k)).

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

Original entry on oeis.org

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

A251820 Numbers n for which the symmetric representation of sigma(n) has at least 3 parts, all having the same area.

Original entry on oeis.org

15, 5950

Offset: 1

Hartmut F. W. Hoft, Dec 09 2014

a(3) > 36000000.

Also intersection of A241558 and A241559 (minimum = maximum) minus the union of A238443 and A239929 (number of parts <= 2).

			The parts of the symmetric representations of sigma(15) and sigma(5950) are {8, 8, 8} and {4464, 4464, 4464}, respectively, so a(1) = 15 and a(2) = 5950.
From _Omar E. Pol_, Dec 09 2014: (Start)
Illustration of the symmetric representation of sigma(15) = 8 + 8 + 8 = 24 in the first quadrant:
.
.  _ _ _ _ _ _ _ _ 8
. |_ _ _ _ _ _ _ _|
.                 |
.                 |_ _
.                 |_  |_ 8
.                   |   |_
.                   |_ _  |
.                       |_|_ _ _ 8
.                             | |
.                             | |
.                             | |
.                             | |
.                             | |
.                             | |
.                             | |
.                             |_|
.
The three parts have the same area.
(End)

Cf. A000203, A237593, A237270, A238443, A239663, A241558, A241559.

(* T[], row[], cD[] & tD[] are defined in A239663 *)
a251820[n_] := Module[{pT = T[n, 1], cT, cL, cW = 0, cR = 0, sects = {}, j = 1, r = row[n], test = True}, While[test && j <= r, cT = T[n, j+1]; cL = pT - cT; cW += (-1)^(j+1) * tD[n, j]; If[cW == 0 && cR != 0, AppendTo[sects, cR]; cR = 0; If[Min[sects] != Max[sects], test = False], cR += cL * cW]; pT = cT; j++]; If[cW != 0, AppendTo[sects, 2 * cR - cW]]; Min[sects] == Max[sects] && Length[sects] > 1]
Select[Range[50000], a251820] (* data *)

A347979 a(n) is the smallest even number k whose symmetric representation of sigma(k) has maximum width n.

Original entry on oeis.org

2, 6, 60, 120, 360, 840, 3360, 2520, 5040, 10080, 15120, 32760, 27720, 50400, 98280, 83160, 110880, 138600, 221760, 277200, 332640, 360360, 554400, 960960, 831600, 942480, 720720, 2217600, 1965600, 1441440

Offset: 1

Hartmut F. W. Hoft, Sep 22 2021

For the 30 known terms the symmetric representation of sigma consists of a single part, i.e., this is a subsequence of A174973 = A238443.
The sequence is not increasing with the maximum width of the symmetric representation of sigma.
Also a(33) = 2162160 is the only further number in the sequence less than 2500000.

			The pattern of maximum widths within the single part of the symmetric representation of sigma for the first four numbers in the sequence is:
  a(n) parts successive widths
    2:   1           1
    6:   1         1 2 1
   60:   1     1 2 3 2 3 2 1
  120:   1     1 2 3 4 3 2 1

Cf. A174973, A237048, A237270, A237271, A237591, A237593, A238443, A249351 (widths), A250070, A262045, A341969, A341970, A341971, A347980.

a262045[n_] := Module[{a=Accumulate[Map[If[Mod[n - # (#+1)/2, #]==0, (-1)^(#+1), 0] &, Range[Floor[(Sqrt[8n+1]-1)/2]]]]}, Join[a, Reverse[a]]]
a347979[n_, mw_] := Module[{list=Table[0, mw], i, v}, For[i=2, i<=n, i+=2, v=Max[a262045[i]]; If [list[[v]]==0, list[[v]]=i]]; list]
a347979[2500000, 33] (* computes a(1..30), a(33); a(31..32) > 2500000 *)

It appears that a(n) = A250070(n) if n >= 2.

A348171 Square array read by upward antidiagonals in which T(w,p) is the smallest number k whose symmetric representation of sigma(k) consists of p parts with maximum width w occurring in at least one of its p parts.

Original entry on oeis.org

1, 6, 3, 60, 78, 9, 120, 7620, 15, 21, 360, 28920, 315, 75, 81, 840, 261720, 1326, 495, 63, 147, 3360, 1422120, 3465, 22542, 525, 189, 729, 2520, 22622880, 17325, 44574, 5005, 1275, 357, 903, 5040, 12728520, 45045, 199578, 6435, 16575, 1287, 1197, 3025, 10080, 50858640, 51975, 7734558, 34034, 131835, 2145, 3861, 2499, 6875

Offset: 1

Hartmut F. W. Hoft, Oct 04 2021

The first row of the table below is A318843 and the first column is A250070.
T(1,k+1) <= 3^k, for all k>=0, since for k=2j the (j+1)-st part in the symmetric representation of sigma(3^k) extends across the diagonal, and for k=2j+1 the (j+1)-st part is completed before the diagonal.
The data computed so far for a partially filled table of 15 rows and 15 columns, show that all rows, all columns (except column 4 for n <= 6 *10^7), and the diagonal are nonmonotonic.

			The 10x10 section of the table with dashes indicating values greater than 6*10^7; rows w denote the maximum width and columns p the number of parts in the symmetric representation of sigma(T(w,p)).
w\p | 1     2        3      4       5       6       7       8        9   ...
----------------------------------------------------------------------------
  1 | 1     3        9      21      81      147     729     903      3025
  2 | 6     78       15     75      63      189     357     1197     2499
  3 | 60    7620     315    495     525     1275    1287    3861     3591
  4 | 120   28920    1326   22542   5005    16575   2145    29325    11583
  5 | 360   261720   3465   44574   6435    131835  76125   24225    82593
  6 | 840   1422120  17325  199578  34034   83655   196707  468027   62985
  7 | 3360  22622880 45045  7734558 153153  442442  314925  1108965  471975
  8 | 2520  12728520 51975     -    205275  2067065 1429275 2359875  557175
  9 | 5040  50858640 225225    -    646646  2863718 2395197 5353725  2785875
  10| 10080    -     405405    -    1990989 2124694 6500375 36535499 7753875
   ...
The symmetric representation of sigma for T(2,3) = 15 consists of the three parts (8, 8, 8) of maximum widths (1, 2, 1), and that of T(3,3) = 315 consists of the three parts (158, 308, 158) of maximum widths (1, 3, 1).

Cf. A237048, A237270, A237271, A237591, A237593, A238443, A239663, A249223, A250070, A262045, A318843, A341969, A341970, A341971, A347979, A347980, A348142.

(* function a341969 is defined in A341969 *)
a348171[n_,  {w_, p_}] := Module[{list=Table[0, {i, w}, {j, p}], k, s, c, u}, For[k=1, k<=n, k++, s=Map[Max, Select[SplitBy[a341969[k], # != 0 &], #[[1]] != 0 &]]; c = Length[s]; u = Max[s]; If[u<=w && c<=p, If[list[[u, c]] == 0, list[[u, c]] = k ]]]; list]
table=a348171[60000000, {15, 15}] (* 15x15 table; very long computation time *)
p[n_] := n-row[n-1](row[n-1]+1)/2
w[n_] := row[n-1]-p[n]+2
Map[table[[w[#], p[#]]]&, Range[55]] (* sequence data *)

a((w+p-2)(w+p-1)/2 + p) = T(w,p), for all w, p >= 1.

T(w(n), p(n)) = a(n), for all n >= 1, where p(n) = n - r(n-1) * (r(n-1) + 1)/2, w(n) = r(n-1) - p(n) + 2, and r(n) = floor((sqrt(8*n+1) - 1)/2).

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.

A352030 Numbers n for which every part of the symmetric representation of sigma(n) has maximum width 2.

Original entry on oeis.org

6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 66, 78, 80, 88, 96, 100, 102, 104, 108, 112, 114, 132, 138, 150, 156, 160, 162, 174, 176, 186, 192, 196, 200, 204, 208, 220, 222, 224, 228, 234, 246, 258, 260, 272, 276, 282, 294, 304, 306, 308, 318, 320, 324, 340, 342, 348

Offset: 1

Hartmut F. W. Hoft, Mar 04 2022

nonn

All numbers in the sequence are even, since for an odd number n the first part of the symmetric representation of sigma(n) has length (n+1)/2 and width 1.
This sequence appears to be a subsequence of A005835 (checked through 30000). It contains all even perfect numbers (A000396) since their symmetric representation consists of a single part that has maximum width 2 only at the diagonal (A174973 and A238443) as represented here by the regular expression DD (see below).
The first pseudoperfect number not in this sequence is 60 since its symmetric representation of sigma consists of a single part with maximum width 3 in 2 symmetric locations.
The patterns of widths in the parts up to the diagonal of the symmetric representation of sigma can be expressed in terms of regular expressions based on the positions of the 1's in the rows of the triangle for sequence A237048, letter D (E) stands for an odd (even) position of a 1. For example, the regular expressions DD and DDED represent 1 part, DDEE and DDEDEE represent 2 parts, etc.
The regular expression for any single complete part, i.e., one not crossing the diagonal, of the symmetric representation of sigma has the form DD...EE and satisfies the inequalities 1 <= #D - #E <= 2 for any proper initial segment of the regular expression. The forms for the two regular expressions when a part crosses the diagonal are similar (see also the comments in A249223).
If n = 2^m * q with m > 0 and q > 1 odd then the numeric requirements corresponding, for example, to the regular expression DDEDEE for the symmetric representation of sigma(n) are that n has 6 odd divisors that are represented by the 6 inequalities 1 = d_1 < d_2 < 2^(m+1) * d_1 < d_3 < 2^(m+1) * d_2 < 2^(m+1) * d_3 <= floor((sqrt(8*n + 1) - 1)/2).

			a(1) = 6 and a(3) = 18 each consist of a single part with respective width patterns 1 2 1 and 1 2 1 2 1 for their entire symmetric representation of sigma, i.e. their respective regular expressions are DD and DDE.
a(15) = 78 is the first number whose entire symmetric representation of sigma consists of 2 parts with width pattern 1 2 1 0 1 2 1, i.e., its regular expression is DDEE (up to the diagonal).
a(158) = 1014 is the first with 3 parts and a(1650) = 12246 the first with 4 parts in their symmetric representation of sigma.

Cf. A000396, A005835, A174973, A235791, A236104, A237048, A237591, A237593, A238443, A249223, A250068, A262045, A279693.

(* Function a237048[ ] is defined in A237048 *)
t237048ToString[n_] := StringJoin[Map[If[OddQ[#], "D", "E"]&, Flatten[Position[a237048[n], 1]]]]
patternTestQ[s_] := StringMatchQ[s, RegularExpression["(DD(ED)*EE)+|(DD(ED)*EE)*DD(ED)*E|(DD(ED)*EE)*DD(ED)*"]]
a352030[n_] := Select[Range[n], patternTestQ[t237048ToString[#]]&]
a352030[350]

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.

A320537 Square array read by antidiagonals in which T(n,k) is the n-th even number j with the property that the symmetric representation of sigma(j) has k parts.

Original entry on oeis.org

2, 4, 10, 6, 14, 50, 8, 22, 70, 230, 12, 26, 98, 250, 1150, 16, 34, 110, 290, 1250, 5050, 18, 38, 130, 310, 1450, 5150, 22310, 20, 44, 154, 370, 1550, 5290, 23230, 106030, 24, 46, 170, 406, 1850, 5350, 23690, 106490, 510050, 28, 52, 182, 410, 2030, 5450, 24610, 107410, 513130, 2065450

Offset: 1

Omar E. Pol, Oct 15 2018

This is a permutation of the positive even numbers (A299174).
The union of all odd-indexed columns gives A319796, the even numbers in A071562.
The union of all even-indexed columns gives A319802, the even numbers in A071561.

			From _Hartmut F. W. Hoft_, Oct 06 2021: (Start)
The 10x10 section of table T(n,k):
(Table with first 20 terms from _Omar E. Pol_)
------------------------------------------------------------------
n\k | 1   2   3    4    5     6     7      8       9       10  ...
------------------------------------------------------------------
  1 | 2   10  50   230  1150  5050  22310  106030  510050  2065450
  2 | 4   14  70   250  1250  5150  23230  106490  513130  2115950
  3 | 6   22  98   290  1450  5290  23690  107410  520150  2126050
  4 | 8   26  110  310  1550  5350  24610  110170  530150  2157850
  5 | 12  34  130  370  1850  5450  25070  112010  530450  2164070
  6 | 16  38  154  406  2030  5650  25250  112930  532450  2168150
  7 | 18  44  170  410  2050  5750  25750  114770  534290  2176550
  8 | 20  46  182  430  2150  6250  25990  115690  537050  2186650
  9 | 24  52  190  434  2170  6350  26450  116150  540350  2216950
  10| 28  58  238  470  2350  6550  26750  117070  544870  2219650
   ... (End)

Row 1 is A320521.
Column 1 gives A174973 = A238443, without the 1.
Column 2 gives A244894.
Cf. A000203, A071561, A071562, A236104, A237270, A237271, A237593, A238443, A239663, A239665, A239929, A240062, A245092, A262626, A299174, A319796, A319802, A341969, A341970, A341971, A346969, A348171.

(* function a341969 is defined in A341969 *)
sArray[b_, pMax_] := Module[{list=Table[{}, pMax], i, p}, For[i=2, i<=b, i+=2, p=Length[Select[SplitBy[a341969[i], #!=0&], #[[1]]!=0&]]; If[p<=pMax&&Length[list[[p]]]Hartmut F. W. Hoft, Oct 06 2021 *)

Terms a(21) and beyond from Hartmut F. W. Hoft, Oct 06 2021

A320048 One half of composite numbers k with the property that the symmetric representation of sigma(k) has two parts.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 67, 68, 69, 71, 73, 74, 76, 79, 82, 83, 86, 87, 89, 92, 93, 94, 97, 101, 103, 106, 107, 109, 111, 113, 116, 118, 122, 123, 124, 127, 129, 131, 134, 137, 139, 141, 142, 146, 148, 149, 151, 157, 158, 159, 163, 164

Offset: 1

Omar E. Pol, Oct 04 2018

nonn

Also, even numbers of A239929 divided by two.

First differs from A101550 at a(51). - R. J. Mathar, Oct 04 2018

			5 is in the sequence because 10 is a composite number, and the symmetric representation of sigma(10) = 18 has two parts (as shown below), and 10/2 = 5.
.
.     _ _ _ _ _ _ 9
.    |_ _ _ _ _  |
.              | |_
.              |_ _|_
.                  | |_ _ 9
.                  |_ _  |
.                      | |
.                      | |
.                      | |
.                      | |
.                      |_|
.

Cf. A101550, A237271 (number of parts), A237270, A237593, A238443, A238524, A239929 (two parts), A239660, A239929, A239932, A239934, A240062 (k parts), A244894, A245092, A262626, A280107 (four parts).

a(n) = A244894(n)/2.

cp's OEIS Frontend

A244579 Numbers k with the property that the number of parts in the symmetric representation of sigma(k) equals the number of divisors of k.

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

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A251820 Numbers n for which the symmetric representation of sigma(n) has at least 3 parts, all having the same area.

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A347979 a(n) is the smallest even number k whose symmetric representation of sigma(k) has maximum width n.

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A348171 Square array read by upward antidiagonals in which T(w,p) is the smallest number k whose symmetric representation of sigma(k) consists of p parts with maximum width w occurring in at least one of its p parts.

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

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A352030 Numbers n for which every part of the symmetric representation of sigma(n) has maximum width 2.

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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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A320537 Square array read by antidiagonals in which T(n,k) is the n-th even number j with the property that the symmetric representation of sigma(j) has k parts.

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