A239663 -id:A239663 - 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)).

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 *)

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).

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

A320521 a(n) is the smallest even number k such that the symmetric representation of sigma(k) has n parts.

Original entry on oeis.org

2, 10, 50, 230, 1150, 5050, 22310, 106030, 510050, 2065450, 10236350

Offset: 1

Omar E. Pol, Oct 14 2018

It appears that a(n) = 2 * q where q is odd and that the symmetric representation of sigma(a(n)/2) has the same number of parts as that for a(n). Number a(12) > 15000000. - Hartmut F. W. Hoft, Sep 22 2021

			a(1) = 2 because the second row of A237593 is [2, 2], and the first row of the same triangle is [1, 1], therefore between both symmetric Dyck paths there is only one part: [3], equaling the sum of the divisors of 2: 1 + 2 = 3. See below:
.
.     _ _ 3
.    |_  |
.      |_|
.
.
a(2) = 10 because the 10th row of A237593 is [6, 2, 1, 1, 1, 1, 2, 6], and the 9th row of the same triangle is [5, 2, 2, 2, 2, 5], therefore between both symmetric Dyck paths there are two parts: [9, 9]. Also there are no even numbers k < 10 whose symmetric representation of sigma(k) has two parts. Note that the sum of these parts is 9 + 9 = 18, equaling the sum of the divisors of 10: 1 + 2 + 5 + 10 = 18. See below:
.
.     _ _ _ _ _ _ 9
.    |_ _ _ _ _  |
.              | |_
.              |_ _|_
.                  | |_ _ 9
.                  |_ _  |
.                      | |
.                      | |
.                      | |
.                      | |
.                      |_|
.
a(3) = 50 because the 50th row of A237593 is [26, 9, 4, 3, 3, 1, 2, 1, 1, 1, 1, 2, 1, 3, 3, 4, 9, 26], and the 49th row of the same triangle is [25, 9, 5, 3, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 3, 5, 9, 25], therefore between both symmetric Dyck paths there are three parts: [39, 15, 39]. Also there are no even numbers k < 50 whose symmetric representation of sigma(k) has three parts. Note that the sum of these parts is 39 + 15 + 39 = 93, equaling the sum of the divisors of 50: 1 + 2 + 5 + 10 + 25 + 50 = 93. (The diagram of the symmetric representation of sigma(50) = 93 is too large to include.)

Row 1 of A320537.
Cf. A237270 (the parts), A237271 (number of parts), A174973 (one part), A239929 (two parts), A279102 (three parts), A280107 (four parts), A320066 (five parts), A320511 (six parts).
Cf. A000203, A018262, A005843, A196020, A235791, A236104, A237048, A237591, A237593, A239663, A239665, A240062, A245092, A262626, A296508.
Cf. A250070, A262045, A341969, A341970, A341971, A347979.

(* support functions are defined in A341969, A341970 & A341971 *)
a320521[n_, len_] := Module[{list=Table[0, len], i, v}, For[i=2, i<=n, i+=2, v=Count[a341969[i], 0]+1;If[list[[v]]==0, list[[v]]=i]]; list]
a320521[15000000,11] (* Hartmut F. W. Hoft, Sep 22 2021 *)

a(6)-a(11) from Hartmut F. W. Hoft, Sep 22 2021

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

A351903 a(n) is the smallest number k such that the symmetric representation of sigma(k) has at least one part of size n.

Original entry on oeis.org

1, 3, 2, 7, 9, 11, 4, 15, 10, 19, 21, 6, 25, 27, 8, 31, 33, 22, 37, 39, 26, 43, 45, 47, 49, 51, 34, 12, 57, 38, 16, 45, 65, 67, 69, 46, 73, 75, 18, 79, 81, 20, 85, 87, 58, 91, 93, 62, 52, 99, 101, 103, 105, 70, 109, 28, 74, 115, 117, 24, 121, 123, 32, 127, 129, 86, 133, 135, 137, 76

Offset: 1

Omar E. Pol, Feb 25 2022

nonn

Conjecture 1: there are infinitely many pairs of the form a(x) = y; a(y) = x (see examples).
First differs from A351904 at a(11).
From Hartmut F. W. Hoft, Jun 10 2024: (Start)
For numbers less than or equal to a(2^20), (2^k, 2^(k+1) - 1), 0 <= k <= 19, are the only pairs satisfying a(a(x)) = x; the triple (36, 46, 91) is the only one satisfying a(a(a(x))) = x, and there are no proper order 4 quadruples and no order 5 quintuples, apart from fixed point 1.
Conjecture 2: Only the pairs x = 2^k and y = 2^(k+1) - 1, k >= 0, satisfy a(x) = y and a(y) = x.
A repeated number d in this sequence determines a pair of distinct indices u and v such that d = a(u) = a(v). This means that d is the smallest number for which parts of sizes u and v occur in the symmetric representation of sigma(d), SRS(d). There are 5507 such pairs less than a(2^20). (End)

			For n = 11 we have that 21 is the smallest number k with at least one part 11 in the symmetric representation of sigma(k), so a(11) = 21.
The symmetric representation of sigma(21) in the first quadrant looks like this:
.
   _ _ _ _ _ _ _ _ _ _ _ 11
  |_ _ _ _ _ _ _ _ _ _ _|
                        |
                        |
                        |_ _ _
                        |_ _  |_ 5
                            |_ _|_
                                | |_ 5
                                |_  |
                                  | |
                                  |_|_ _ _ _ 11
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          |_|
.
For n = 12 we have that 6 is the smallest number k with at least one part 12 in the symmetric representation of sigma(k), so a(12) = 6.
The symmetric representation of sigma(6) in the first quadrant looks like this:
.
   _ _ _ _
  |_ _ _  |_
        |   |_ 12
        |_ _  |
            | |
            | |
            |_|
.
Some pairs of the form a(x) = y; a(y) = x:
   a(2) =  3;   a(3) =  2.
   a(4) =  7;   a(7) =  4.
   a(8) = 15;  a(15) =  8.
  a(16) = 31;  a(31) = 16.
.
The first three repeated terms are 45 = a(23) = a(32), 135 = a(68) = a(104) and 225 = a(113) = a(177), - _Hartmut F. W. Hoft_, Jun 10 2024

Hartmut F. W. Hoft, Table of n, a(n) for n = 1..1000

Cf. A351904 (Analog for subparts).

Cf. A000079, A000203, A000225, A196020, A235791, A236104, A237270 (parts), A237271, A237591, A237593, A239660, A239663, A239931, A239932, A239933, A239934, A241558, A245092, A262626.

(* a237270[ ] implements improved computing time for A237270 *)
a237591[n_, k_] := Map[Ceiling[(n+1)/#-(#+1)/2]-Ceiling[(n+1)/(#+1)-(#+2)/2]&, Range[k]]
a249223[n_, k_] := FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, k]]
row[n_] := Floor[(Sqrt[8*n+1]-1)/2]
a237270[n_] := Module[{lw=Transpose[{a237591[n, row[n]], a249223[n, row[n]]}], diag, sL, pL}, diag=Last[lw][[2]]; sL=Split[lw, #[[2]]!=0&]; pL=Map[Apply[Dot, Transpose[#]]&, Select[sL, #[[1, 2]]!=0&]]; If[diag==0, Join[pL, Reverse[pL]], If[Length[pL]>1, Join[Most[pL], {2Last[pL]-diag}, Reverse[Most[pL]]], 2pL-diag]]]
a351903[n_] := Module[{list=Table[0, n], count0=n, i=1}, While[count0>0, Map[If[list[[#]]==0, list[[#]]=i; count0--]&, Select[a237270[i], #<=n&]]; i++]; list]
a351903[70] (* Hartmut F. W. Hoft, Jun 10 2024 *)

a(n) = min( k : exists 1 <= j <= |SRS(k)|, SRS(k)[j] = n ) where |SRS(k)| denotes the number of parts in SRS(k) and SRS(k)[j] its j-th part. - Hartmut F. W. Hoft, Jun 10 2024

a(17)-a(70) from Hartmut F. W. Hoft, Jun 10 2024

A266000 Numbers k such that the symmetric representation of sigma(k) has at least two parts of distinct size.

Original entry on oeis.org

9, 21, 25, 27, 33, 35, 39, 45, 49, 50, 51, 55, 57, 63, 65, 69, 70, 75, 77, 81, 85, 87, 91, 93, 95, 98, 99, 105, 110, 111, 115, 117, 119, 121, 123, 125, 129, 130, 133, 135, 141, 143, 145, 147, 153, 154, 155, 159, 161, 165, 169, 170, 171, 175, 177, 182, 183, 185, 187, 189, 190, 195

Offset: 1

Omar E. Pol, Dec 19 2015

nonn

In other words: numbers k such that the symmetric representation of sigma(k) has at least two parts with distinct number of cells.
For more information about the symmetric representation of sigma see A237270 and A237593.
When the symmetric representation of sigma of m, SRS(m), consists of 2n-1 or 2n parts, n>=1, then at most n parts can be of distinct sizes. For the published terms in A239663, SRS(A239663(n)) consists of n parts representing ceiling(n/2) parts of distinct sizes, n>=1. Only two numbers m are known, 15 and 5950 in A251820, for which SRS(m) consists of n parts of less than ceiling(n/2) distinct sizes. - Hartmut F. W. Hoft, Jan 11 2025

			The symmetric representation of sigma(9) = 13 in the first quadrant looks like this:
y
.
._ _ _ _ _ 5
|_ _ _ _ _|
.         |_ _ 3
.         |_  |
.           |_|_ _ 5
.               | |
.               | |
.               | |
.               | |
. . . . . . . . |_| . . x
.
There are three parts: 5 + 3 + 5 = 13, so 9 is in the sequence because the structure contains at least two parts of distinct size.
From _Hartmut F. W. Hoft_, Jan 11 2025: (Start)
SRS(a(1)) = SRS(A239663(3)) = SRS(9) = { 5, 3, 5 } is the smallest with 2 parts of distinct sizes.
SRS(a(14)) = SRS(A239663(5)) = SRS(63) = { 32, 12, 16, 12, 32 } is the smallest with 3 parts of distinct sizes.
SRS(a(127)) = SRS(A239663(7)) = SRS(357) = { 179, 61, 29, 38, 29, 61, 179 } is the smallest with 4 parts of distinct sizes. (End)

Complement of A265999.
Cf. A000203, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A245092, A262626.
Cf. A239663, A251820, A377654.

(* Function partsSRS[ ] is defined in A377654 *)
a266000[n_] := Select[Range[n], Length[Union[partsSRS[#]]]>=2&]
a266000[200] (* Hartmut F. W. Hoft, Jan 11 2025 *)

Extended from a(37) to a(62) by Hartmut F. W. Hoft, Jan 11 2025

A346969 1 together with the square array T(n,k) read by upward antidiagonals in which T(n, k), n >= 1, is the n-th odd number j >= 3 such that the symmetric representation of sigma of j has k >= 2 parts.

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 21, 11, 25, 27, 63, 13, 35, 33, 81, 147, 17, 45, 39, 99, 171, 357, 19, 49, 51, 117, 189, 399, 903, 23, 77, 55, 153, 207, 441, 987, 2499, 29, 91, 57, 165, 243, 483, 1029, 2709, 6069, 31, 121, 65, 195, 261, 513, 1113, 2793, 6321, 13915, 37, 135, 69, 231, 275, 567, 1197, 2961, 6325, 14847, 29095

Offset: 1

Hartmut F. W. Hoft, Oct 06 2021

This sequence is a permutation of the odd positive integers.

The first row of table T(n,k) preceded by a(1) = 1 is A239663; the first column is the sequence A065091 of odd primes; the second column contains the squares of the odd primes as a subsequence (see also A247687).

			The 10x10 initial submatrix of table T(n,k):
n\k | 2   3    4    5     6     7     8      9      10     11  ...
------------------------------------------------------------------
  1 | 3   9    21   63    147   357   903    2499   6069   13915
  2 | 5   15   27   81    171   399   987    2709   6321   14847
  3 | 7   25   33   99    189   441   1029   2793   6325   15125
  4 | 11  35   39   117   207   483   1113   2961   6783   15141
  5 | 13  45   51   153   243   513   1197   3025   6875   15351
  6 | 17  49   55   165   261   567   1239   3087   6909   15729
  7 | 19  77   57   195   275   609   1265   3249   7011   16023
  8 | 23  91   65   231   279   621   1281   3339   7203   16611
  9 | 29  121  69   255   297   651   1375   3381   7353   16779
  10| 31  135  75   273   333   729   1407   3591   7581   17157
   ...
a(9) = 25 = T(3,3) since only 9 and 15 are smaller odd numbers whose symmetric representation of sigma consists of three parts. All 3 parts of the symmetric representation of sigma for 9 and for 25 have width 1 while the center part for that of 15 has width 2.

Cf. A065091, A237270, A237271, A237593, A239663, A247687, A320537, A341969, A341970, A341971, A348171.

(* function a341969 is defined in A341969 *)
sArray[b_, pMax_] := Module[{list=Table[{}, pMax+1], i, p}, For[i=3, i<=b, i+=2, p=Length[Select[SplitBy[a341969[i], #!=0&], #[[1]]!=0&]]; If[p<=pMax+1&&Length[list[[p]]]

A348142 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 everyone of its p parts.

Original entry on oeis.org

1, 6, 3, 60, 78, 9, 120, 7620, 1014, 21, 360, 28920, 967740, 12246, 81, 840, 261720, 6969720, 116136420, 171366, 147, 3360, 1422120

Offset: 1

Hartmut F. W. Hoft, Oct 04 2021

It appears that the first row is A318843 and that the first column is A250070.
Columns 1 and 2 both are identical with those of the table in A348171 and row 1 is identical with that of A348171.
In the remainder of the 7th antidiagonal a(24..26) > 120*10^6, a(27) = 1922622, and a(28) = 903.

			The 10x8 section of the table T(w,p) with dashes indicating values greater than 120*10^6; rows w denote the common maximum width in all parts 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  ...
--------------------------------------------------------------------------
  1 | 1     3         9        21        81       147     729      903
  2 | 6     78        1014     12246     171366   1922622 28960854  -
  3 | 60    7620      967740   116136420   -         -       -
  4 | 120   28920     6969720    -
  5 | 360   261720      -
  6 | 840   1422120     -
  7 | 3360  22622880    -
  8 | 2520  12728520    -
  9 | 5040  50858640    -
  10| 10080    -
   ...
The symmetric representation of sigma for T(2,2) = 78 consists of the two parts (84, 84) of maximum widths (2, 2), and that of T(2,3) = 1014 consists of the three parts (1020, 156, 1020) of maximum widths (2, 2, 2).

Cf. A237048, A237270, A237271, A237591, A237593, A238443, A239663, A249223, A250070, A262045, A318843, A341969, A341970, A341971, A347979, A347980, A348171.

(* function a341969 is defined in A341969 *)
a348142[n_, {w_, p_}] := Module[{list=Table[0, {i, w}, {j, p}], k, s, c, u}, Monitor[For[k=1, k<=n, k++, s=Map[Max, Select[SplitBy[a341969[k], #!=0&], #[[1]]!=0&]]; c=Length[s]; u=Union[s]; If[Length[u]==1&&u[[1]]<=w&&c<=p, If[list[[u[[1]], c]]==0, list[[u[[1]], c]]=k]]], list]; list]
table=a348142[120000000, {10, 10}] (* 10x10 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[23]] (* 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).

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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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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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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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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A320521 a(n) is the smallest even number k such that the symmetric representation of sigma(k) has n parts.

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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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A351903 a(n) is the smallest number k such that the symmetric representation of sigma(k) has at least one part of size n.

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A266000 Numbers k such that the symmetric representation of sigma(k) has at least two parts of distinct size.

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