A318843 -id:A318843 - cp's OEIS frontend

A241010 Numbers n with the property that the number of parts in the symmetric representation of sigma(n) is odd, and that all parts have width 1.

1, 2, 4, 8, 9, 16, 25, 32, 49, 50, 64, 81, 98, 121, 128, 169, 242, 256, 289, 338, 361, 484, 512, 529, 578, 625, 676, 722, 729, 841, 961, 1024, 1058, 1156, 1250, 1369, 1444, 1681, 1682, 1849, 1922, 2048, 2116, 2209, 2312, 2401, 2738, 2809, 2888, 3025, 3249, 3362, 3364, 3481, 3698, 3721, 3844

Offset: 1

Hartmut F. W. Hoft, Aug 07 2014

nonn

The first eight entries in A071562 but not in this sequence are 6, 12, 15, 18, 20, 24, 28 & 30.
The first eight entries in A238443 but not in this sequence are 6, 12, 18, 20, 24, 28, 30 & 36.
The union of A241008 and of this sequence equals A174905 (for a proof see link in A174905).
Let n = 2^m * product(p_i^e_i, i=1,...,k) = 2^m * q with m >= 0, k >= 0, 2 < p_1, ...< p_k primes and e_i >= 1, for all 1 <= i <= k. For each number n in this sequence all e_i are even, and for any two odd divisors f < g of n, 2^(m+1) * f < g. The sum of the areas of the regions r(n, z) equals sigma(n). For a proof of the characterization and the formula see the theorem in the link below.
Numbers 3025 = 5^2 * 11^2 and 510050 = 2^1 * 5^2 * 101^2 are the smallest odd and even numbers, respectively, in the sequence with two distinct odd prime factors.
Among the 706 numbers in the sequence less than 1000000 (see link to the table) there are 143 that have two different odd prime factors, but none with three. All numbers with three different odd prime factors are larger than 500000000. Number 4450891225 = 5^2 * 11^2 * 1213^2 is in the sequence, but may not be the smallest one with three different odd prime factors. Note that 1213 is the first prime that extends the list of divisors of 3025 while preserving the property for numbers in this sequence.
The subsequence of numbers n = 2^(k-1) * p^2 satisfying the constraints above is A247687.
n = 3^(2*h) = 9^h = A001019(h), h>=0, is the smallest number for which the symmetric representation of sigma(n) has 2*h+1 regions of width one, for example for h = 1, 2, 3 and 5, but not for h = 4 in which case 3025 = 5^2 * 11^2 < 3^8 = 6561 is the smallest (see A318843). [Comment corrected by Hartmut F. W. Hoft, Sep 04 2018]
Computations using this characterization are more efficient than those of Dyck paths for the symmetric representations of sigma(n), e.g., the Mathematica code below.

			This irregular triangle presents in each column those elements of the sequence that have the same factor of a power of 2.
  row/col      2^0    2^1   2^2   2^3    2^4    2^5  ...
   2^k:          1      2     4     8     16     32  ...
   3^2:          9
   5^2:         25     50
   7^2:         49     98
   3^4:         81
  11^2:        121    242   484
  13^2:        169    338   676
  17^2:        289    578  1156  2312
  19^2:        361    722  1444  2888
  23^2:        529   1058  2116  4232
   5^4:        625   1250
   3^6:        729
  29^2:        841   1682  3364  6728
  31^2:        961   1922  3844  7688
  37^2:       1369   2738  5476 10952 21904
  41^2:       1681   3362  6724 13448 26896
  43^2:       1849   3698  7396 14792 29584
  47^2:       2209   4418  8836 17672 35344
   7^4:       2401   4802
  53^2:       2809   5618 11236 22472 44944
  5^2*11^2:   3025
  3^2*19^2:   3249
  59^2:       3481   6962 13924 27848 55696
  61^2:       3721   7442 14884 29768 59536
  67^2:       4489   8978 17956 35912 71824 143648
  3^2*23^2:   4761
  71^2:       5041
  ...
  5^2*101^2:225025 510050
  ...
Number 3025 = 5^2 * 11^2 is in the sequence since its divisors are 1, 5, 11, 25, 55, 121, 275, 605 and 3025. Number 6050 = 2^1 * 5^2 * 11^2 is not in the sequence since 2^2 * 5 > 11 while 5 < 11.
Number 510050 = 2^1 * 5^2 * 101^2 is in the sequence since its 9 odd divisors 1, 5, 25, 101, 505, 2525, 10201, 51005 and 225025 are separated by factors larger than 2^2. The areas of its 9 regions are 382539, 76515, 15339, 3939, 1515, 3939, 15339, 76515 and 382539. However, 2^2 * 5^2 * 101^2 is not in the sequence.
The first row is A000079.
The rows, except the first, are indexed by products of even powers of the odd primes satisfying the property, sorted in increasing order.
The first column is a subsequence of A244579.
A row labeled p^(2*h), h>=1 and p>=3 with p = A000040(n), has A098388(n) entries.
Starting with the second column, dividing the entries of a column by 2 creates a proper subsequence of the prior column.
See A259417 for references to other sequences of even powers of odd primes that are subsequences of column 1.
The first entry greater than 16 in column labeled 2^4 is 21904 since 37 is the first prime larger than 2^5. The rightmost entry in the row labeled 19^2 is 2888 in the column labeled 2^3 since 2^4 < 19 < 2^5.

Hartmut F. W. Hoft, Table of n, a(n) for n = 1..563 (values less than 1000000)
Hartmut F. W. Hoft, Proof of characterization theorem
Hartmut F. W. Hoft, Table of n, a(n) for n = 1...706(values less than 1000000)
Hartmut F. W. Hoft, Illustration of the symmetric representation of sigma(a(n)), for n=1..15
Hartmut F. W. Hoft, Proof of property for n and formula for regions of sigma(n)

Cf. A000203, A174905, A236104, A237270 (symmetric representation of sigma(n)), A237271, A237593, A238443, A241008, A071562, A246955, A247687, A250068, A250070, A250071.

Cf. A318843

(* path[n] and a237270[n] are defined in A237270 *)
atmostOneDiagonalsQ[n_] := SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], -1] - path[n-1], 1]]]
Select[Range[1000], atmostOneDiagonalsQ[#] && OddQ[Length[a237270[#]]]&] (* data *)
(* more efficient code based on numeric characterization *)
divisorPairsQ[m_, q_] := Module[{d = Divisors[q]}, Select[2^(m + 1)*Most[d] - Rest[d], # >= 0 &] == {}]
a241010AltQ[n_] := Module[{m, q, p, e}, m=IntegerExponent[n, 2]; q=n/2^m; {p, e} = Transpose[FactorInteger[q]]; q==1||(Select[e, EvenQ]==e && divisorPairsQ[m, q])]
a241010Alt[m_,n_] := Select[Range[m, n], a241010AltQ]
a241010Alt[1,4000] (* data *)

Formula for the z-th region in the symmetric representation of n = 2^m * q in this sequence, 1 <= z <= sigma_0(q) and q odd: r(n, z) = 1/2 * (2^(m+1) - 1) * (d_z + d_(2*x+2-z)) where 1 = d_1 < ... < d_(2*x+1) = q are the odd divisors of n.

More terms and further edited by Hartmut F. W. Hoft, Jun 26 2015 and Jul 02 2015 and corrected Oct 11 2015

A367377 Square array T(n, k), n >= 1, k >= 1, read by antidiagonals, of the least numbers whose symmetric representation of sigma instantiate the unimodal width pattern 1, 2, ..., n, ..., 2, 1 repeated k times separated by instances of width 0.

Original entry on oeis.org

1, 6, 3, 72, 78, 9, 120, 10728, 1014, 21, 5184, 28920, 1598472, 12246, 81, 1440, 53752896, 6969720, 230297976, 171366, 147, 373248, 4157280

Offset: 1

Hartmut F. W. Hoft, Nov 15 2023

The numbers T(n, 1) instantiating a single unimodal pattern of width n form A250071(n). This first column is not increasing since T(5, 1) = 5184 > 1440 = T(6, 1).
The numbers T(1, k) instantiating the repeating unimodal patterns 1, 1, 0, 1, ..., 1, 0, 1, 0, ..., 0, 1, 0, 1, ... of width 1 form A318843(k). This first row is not increasing since T(1, 11) = 59049 > 29095 = T(1, 12).
The rows in the table are infinite since the numbers T(n, 1) * p^(k-1) >= T(n, k), with p the smallest prime greater than 2 * T(n, 1), instantiate the width pattern for T(n, k), though equality need not hold, as T(1, 4) = 21 = 3 * 7 < 1 * 3^3 = 27 demonstrates.
Conjecture 1: None of the rows and columns are increasing.
Conjecture 2: T(n, p) = T(n, 1) * A151800(2*T(n, 1))^(p-1) for n >= 1 and primes p.
Conjecture 3: T(p, q), p and q primes, is a record for its upper left hand rectangle in the table. Only one prime number index generally is not sufficient as the inequality 4157280 = T(6, 2) < 5 * 10^6 < T(5, 2) shows.

			The corner of the table begins:
  --------------------------------------------------------------------
     Pattern   |   once    twice  3 times   4 times  5 times  6 times
  --------------------------------------------------------------------
        1      |      1        3        9        21       81      147
       121     |      6       78     1014     12246   171366  1922622
      12321    |     72    10728  1598472 230297976
     1234321   |    120    28920  6969720
    123454321  |   5184 53752896
   12345654321 |   1440  4157280
  1234567654321| 373248
  ...
T(3, 4) must have 12 odd divisors and as least number must have 2^3 * 3^2 as a factor in order to create the initial width pattern 1 2 3 2 1 0. Therefore, since the next smallest prime larger than 16 * 9 is 149, T(3, 4) is 2^3 * 3^2 * 149^3 or 2^3 * 3^2 * 149 * p for suitable prime p which leads to p = 21467 < 22201 = 149^2.
All other numbers in the table were found by exhaustive computations.

Cf. A235791, A237048, A237270, A237591, A237593, A249223, A250071, A262045, A318843, A341969, A342592, A342594, A342595, A342596.

t249223[n_] := FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, Floor[(Sqrt[8n+1]-1)/2]]] (* row n in triangle of A249223 *)
t262045[n_] := Join[t249223[n], Reverse[t249223[n]]] (* row n in triangle of A262045 *)
widthPattern[n_] := Map[First, Split[t262045[n]]]
umw[n_, k_] := Most[Flatten[Table[Join[Range[n], Range[n-1, 0, -1]], k]]]
a367377[{n_, k_}, b_] := NestWhile[#+1&, 1, #

A357581 Square array read by antidiagonals of numbers whose symmetric representation of sigma consists only of parts that have width 1; column k indicates the number of parts and row n indicates the n-th number in increasing order in each of the columns.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 8, 7, 25, 21, 16, 10, 49, 27, 81, 32, 11, 50, 33, 625, 147, 64, 13, 98, 39, 1250, 171, 729, 128, 14, 121, 51, 2401, 207, 15625, 903, 256, 17, 169, 55, 4802, 243, 31250, 987, 3025, 512, 19, 242, 57, 14641, 261, 117649, 1029, 3249, 6875

Offset: 1

Hartmut F. W. Hoft, Oct 04 2022

This sequence is a permutation of A174905. Numbers in the even numbered columns of the table form A241008 and those in the odd numbered columns form A241010. The first row of the table is A318843.

This sequence is a subsequence of A240062 and each column in this sequence is a subsequence in the respective column of A240062.

			The upper left hand 11 X 11 section of the table for a(n) <= 2*10^7:
     1   2    3   4      5    6         7     8      9     10        11 ...
  ----------------------------------------------------------------------
     1   3    9  21     81  147       729   903   3025   6875     59049
     2   5   25  27    625  171     15625   987   3249   7203   9765625
     4   7   49  33   1250  207     31250  1029   4761  13203  19531250
     8  10   50  39   2401  243    117649  1113   6561  13527       ...
    16  11   98  51   4802  261    235298  1239   7569  14013       ...
    32  13  121  55  14641  275   1771561  1265   8649  14499       ...
    64  14  169  57  28561  279   3543122  1281  12321  14661       ...
   128  17  242  65  29282  333   4826809  1375  14161  15471       ...
   256  19  289  69  57122  363   7086244  1407  15129  15633       ...
   512  22  338  85  58564  369   9653618  1491  16641  15957       ...
  1024  23  361  87  83521  387  19307236  1533  17689  16119       ...
  ...
Each column k > 1 contains odd and even numbers since, e.g., 5^(k-1) and 2 * 5^(k-1) belong to it.
Column 1: A000079, subsequence of A174973 = A238443, and of column 1 in A240062.
Column 2: A246955, subsequence of A239929; 78 is the smallest number not in A246955.
Column 3: A247687, subsequence of A279102; 15 is the smallest number not in A247687.
  Odd numbers in column 3: A001248(k), k > 1.
Column 4: A264102, subsequence of A280107; 75 is the smallest number not in A264102.
Column 5: subsequence of A320066; 63 = A320066(1) is not in column 5.
  Numbers in column 5 have the form 2^k * p^4 with p > 2 prime and 0 <= k < floor(log_2(p)).
  Odd numbers in column 5: A030514(k), k > 1.
Column 6: subsequence of A320511; 189 is the smallest number not in column 6.
  Smallest even number in column 6 is 5050.
Column 7: Numbers have the form 2^k * p^6 with p > 2 prime and 0 <= k < floor(log_2(p)).
  Odd numbers in column 7: A030516(k), k > 1.
Numbers in the column numbered with the n-th prime p_n have the form: 2^k * p^(p_n - 1) with p > 2 prime and 0 <= k < floor(log_2(p_n)).

Cf. A000079, A001248, A030514, A030516, A174905, A174973, A237593, A238443, A239929, A241008, A241010, A246955, A247687, A264102, A279102, A280107, A318843, A320066, A320511, A341969, A341970, A341971.

(* function a341969 and support functions are defined in A341969, A341970 and A341971 *)
width1Table[n_, {r_, c_}] := Module[{k, list=Table[{}, c], wL, wLen, pCount, colLen}, For[k=1, k<=n, k++, wL=a341969[k]; wLen=Length[wL]; pCount=(wLen+1)/2; If[pCount<=c&&Length[list[[pCount]]]=1, j--, vec[[PolygonalNumber[i+j-2]+j]]=arr[[i, j]]]]; vec]
a357581T[n_, r_] := TableForm[width1Table[n, {r, r}]]
a357581[120000, 10] (* sequence data - first 10 antidiagonals *)
a357581T[120000, 10] (* upper left hand 10x10 array *)
a357581T[20000000, 11] (* 11x11 array - very long computation time *)

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).

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).

A338534 a(n) is the smallest number k whose symmetric representation of sigma(k) consists of n regions (or parts) and whose areas are strictly decreasing towards the diagonal.

Original entry on oeis.org

1, 3, 9, 21, 81, 147, 441, 903, 2499, 6069, 15351, 29095, 80367, 153065, 366025, 643885

Offset: 1

Hartmut F. W. Hoft, Nov 01 2020

For a(n) <= 750000 the regions in the symmetric representation of sigma(a(n)) have at most width of 2.

			a(5) = 81 is in this sequence since its symmetric representation of sigma is the first to consist of 5 regions [41, 15, 9, 15, 41] decreasing towards the diagonal.
63 is the only number smaller than 81 with 5 regions, but is not in the sequence since the regions of its symmetric representation of sigma are [32, 12, 16, 12, 32].

Cf. A237048, A237270, A237271, A237593, A239663, A318843.

cd[n_, k_] := If[Divisible[n, k], 1, 0]
legs[n_, len_] := Module[{tL = Map[Ceiling[(n + 1)/# - (# + 1)/2] &, Range[len]]}, tL - Append[Rest[tL], 0]]
a237048[n_, len_] := Map[If[OddQ[#], cd[n, #], cd[n-#/2, #]]&, Range[len]]
widths[n_, len_] := Drop[FoldList[Plus, 0, Map[(-1)^(#+1)&, Range[len]] a237048[n, len]], 1]
regions[n_] := Module[{r=Floor[(Sqrt[8n+1]-1)/2], wL, wM, diag, sL, sLs, regs, lens}, wL=widths[n, r]; wM=Max[wL]; diag=Last[wL]; sL=legs[n, r]wL; sLs=SplitBy[sL, #!=0&]; regs=Select[Map[Fold[Plus, 0, #]&, sLs], #!=0&]; lens=Length[regs]; If[diag==0, {wM, Join[regs, Reverse[regs]]}, {wM, Join[Most[regs], {2Last[regs]-diag}, Reverse[Most[regs]]]}]]
a338534[n_, b_] := Module[{k, r, len, s, list=Table[0, b]}, For[k=1, k<=n, k++, r=Last[regions[k]]; len=Length[r]; s=Take[r, Floor[(len+1)/2]]; If[AllTrue[Most[s]-Rest[s], #>0&]&&list[[len]]==0, list[[len]]=k]]; list]
Take[a338534[750000, 20], 16] (* sequence data *)

A367370 a(k) is the number of different widths patterns in the symmetric representation of sigma for numbers having k odd divisors.

Original entry on oeis.org

1, 2, 3, 6, 5, 16, 7, 40

Offset: 1

Hartmut F. W. Hoft, Dec 05 2023

The width pattern (A341969) of the symmetric representation of sigma for a number with k >= 1 odd divisors has length 2*k - 1.
a(p) = p for any prime number p is realized by the m+1 numbers 3^(p-1), ..., 2^m * 3^(p-1) which contain m+1-p duplicates, where m = floor(log_2(3^(p-1))). Each width pattern first increases to a level 1 <= i <= p and then alternates between i and i-1 up to the diagonal of the symmetric representation of sigma resulting in p distinct patterns.
For some numbers n = 2^m * q, q odd and not prime, that are the least instantiations of a width pattern their odd parts q may not be the least instantiations of a width pattern, examples are 78, 1014, 12246 and 171366 with 4, 6, 8 and 10 odd divisors, respectively (see row 2 of the table in A367377).
Conjecture: a(9) = 28.
The least number instantiating the 28th width pattern, 12345654345654321, is n = 43356672, found in a search up to 5*10^9.
Table of width pattern counts of the symmetric representation of sigma and of all possible symmetric patterns:
# odd divisors  1   2   3   4   5    6    7    8    9   10   11    12
pattern count   1   2   3   6   5   16    7   40   28? >=47  11 >=223
A001405         1   2   3   6  10   20   35   70  126  252  462   924
The 4 symmetric patterns 10123232101, 10123432101, 12101010121 and 12123432121 cannot be instantiated as width patterns of numbers with 6 odd divisors.
30 of the 70 possible symmetric patterns of numbers n = 2^m * q, m>=0 and q odd, with 8 odd divisors cannot be instantiated as width patterns of the symmetric representation of sigma(n) since their sequence of widths contradicts the order of the odd divisors d_i of n and of the numbers 2^(m+1) * d_i and the positions of their corresponding 1's in the rows of the triangle of widths in A249223.

			In the irregular triangle below, row k lists the count and the first occurrences of successive instantiations of the distinct width patterns in the symmetric representation of sigma for numbers with k odd divisors.
# div |count|    first occurrence of distinct width patterns
      |     |    1    2    3     4     5     6      7 .. 11 .. 16 .. 40
-----------------------------------------------------------------------
1     |  1  |    1                                        .     .     .
2     |  2  |    3    6                                   .     .     .
3     |  3  |    9   18   72                              .     .     .
4     |  6  |   15   21   30    60    78   120            .     .     .
5     |  5  |   81  162  648  1296  5184                  .     .     .
6     | 16  |   45   63   75    90   147   150    180    ...  27744   .
7     |  7  |  729 1458 5832 11664 46656 93312 373248     .           .
8     | 40  |  105  135  165   189   210   231    357    ...       203808
9     | 28? |  225  441  450   882   900  1225   1800    ...
10    | >=47|  405  567  810  1134  1377  1539   1620    ...
11    | 11  |59049                 ...               1934917632
The complete sequence of first occurrences of the 11 width patterns for numbers with 11 odd divisors is: 59049, 118098, 472392, 944784, 3779136, 7558272, 30233088, 120932352, 241864704, 967458816, 1934917632.
The column labeled '1' of least occurrences of a width pattern of length 2k-1 is sequence A038547: least number with exactly k odd divisors.

Cf. A235791, A237048, A237270, A237591, A237593, A249223, A250071, A262045, A318843, A341969, A342592, A342594, A342595, A342596, A367377.

t249223[n_] := FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, Floor[(Sqrt[8n+1]-1)/2]]]
(* row n in triangle of A249223 *)
t262045[n_] := Join[t249223[n], Reverse[t249223[n]]] (* row n in triangle of A262045 *)
widthPattern[n_] := Map[First, Split[t262045[n]]]
nOddDivs[n_] := Length[Divisors[NestWhile[#/2&, n, EvenQ[#]&]]]
count[n_, k_] := Length[Union[Map[widthPattern, Select[Range[n], nOddDivs[#]==k&]]]]
(* count of distinct width patterns for numbers with k odd divisors in the range 1 .. n *)

A384704 Triangle T(i, j), 1 <= j <= i, read by rows. T(i, j) is the smallest number k that has i odd divisors and whose symmetric representation of sigma, SRS(k), has j parts; when no such k exists then T(i, j) = -1.

Original entry on oeis.org

1, 6, 3, 18, -1, 9, 30, 78, 15, 21, 162, -1, -1, -1, 81, 90, 666, 45, 75, 63, 147, 1458, -1, -1, -1, -1, -1, 729, 210, 1830, 135, 105, 165, 189, 357, 903, 450, -1, 225, -1, 1225, -1, 441, -1, 3025, 810, 53622, 405, -1, 1377, 1875, 567, 1539, 4779, 6875, 118098, -1, -1, -1, -1, -1, -1, -1, -1, -1, 59049

Offset: 1

Hartmut F. W. Hoft, Jun 07 2025

T(i, j) = -1 for i  >= 1 odd, nonprime, j even with 1 < j < i; also for i prime and all j with 1 < j < i.
The single value T(10, 4) = -1 has been verified; see the conjecture below.
T(i, i) <= 3^(i-1) for all i >=1 . Equality holds for all primes i. T(i, i) = A318843(i), for all i >= 1.
A038547(i) is the smallest number with exactly i odd divisors. Thus odd number A038547(i) occurs in row i of triangle T(i, j) so that A038547 is a subsequence of this sequence. For i prime, A038547(i) = T(i, i). For 4 <= i <= 10^9 nonprime, A038547(i) is in the third column, T(i, 3), except for i=8; furthermore, the first part of SRS(A038547(i)) has width 1 and size (A038547(i)+1)/2.
T(i, 1) <= 2 * 3^(i-1) and it is even for all i >1. Equality holds for all primes i.
T(i, 2) <= 2 * 3^(i/2-1) * p for all even i where p is the smallest prime greater than 4 * 3^(i/2-1). Equality holds when i = 2 * h where h is prime.
The positive numbers in columns 1..6 are subsequences of A174973, A239929, A279102, A280107, A320066, A320511, respectively.
Conjectures:
All entries T(i, j) in columns j >= 3 are odd.
T(i, 1)/2 is odd for all i > 1.
T(i, 1) = 2 * T(i, 3) for all nonprime i > 3, for i = 3, but not for i = 8.
T(i, 2)/2 is odd for all even i > 2.
T(i, 3) = A038547(i) for all nonprime i > 3, except i = 8.
T(2*i, 2*j) = -1 for j >= 2 and all prime i satisfying i >= prime(j+1).
From Omar E. Pol, Jun 08 2025: (Start)
T(i,j) is also the smallest number k whose symmetric representation of sigma(k) has i subparts and j parts, or -1 if no such k exists.
Observations:
At least for i < 12 if i is prime then T(i,1) = 2*T(i,i).
At least for i < 12 if i is prime then all terms in row i are -1's except the first and the last term. (End)

			The first 12 rows of triangle T(i, j):
   i\j      1     2   3   4    5    6    7    8    9   10    11    12
   1:       1
   2:       6     3
   3:      18    -1   9
   4:      30    78  15  21
   5:     162    -1  -1  -1   81
   6:      90   666  45  75   63  147
   7:    1458    -1  -1  -1   -1   -1  729
   8:     210  1830 135 105  165  189  357  903
   9:     450    -1  25  -1 1225   -1  441   -1 3025
  10:     810 53622 405  -1 1377 1875  567 1539 4779 6875
  11:  118098    -1  -1  -1   -1   -1   -1   -1   -1   -1 59049
  12:     630 16290 315 495  525 1071 1287 1197 2499 6069 13915 29095
  ...

Cf. A003056, A038547, A174973, A235791, A237048, A237591, A237593, A239929, A249223, A279102, A279387, A280107, A318843, A320066, A320511, A377654.

(* function partsSRS[ ] is defined in A377654 *)
setupT[d_] := Module[{list=Table[0, {i, d}, {j, i}], s, t}, For[s=1, s<=d, s++, For[t=1, t<=s, t++, If[(OddQ[s]&&Not[PrimeQ[s]]&&EvenQ[t]&&1

cp's OEIS Frontend

A241010 Numbers n with the property that the number of parts in the symmetric representation of sigma(n) is odd, and that all parts have width 1.

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A367377 Square array T(n, k), n >= 1, k >= 1, read by antidiagonals, of the least numbers whose symmetric representation of sigma instantiate the unimodal width pattern 1, 2, ..., n, ..., 2, 1 repeated k times separated by instances of width 0.

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A357581 Square array read by antidiagonals of numbers whose symmetric representation of sigma consists only of parts that have width 1; column k indicates the number of parts and row n indicates the n-th number in increasing order in each of the columns.

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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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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.

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A338534 a(n) is the smallest number k whose symmetric representation of sigma(k) consists of n regions (or parts) and whose areas are strictly decreasing towards the diagonal.

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A367370 a(k) is the number of different widths patterns in the symmetric representation of sigma for numbers having k odd divisors.

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A384704 Triangle T(i, j), 1 <= j <= i, read by rows. T(i, j) is the smallest number k that has i odd divisors and whose symmetric representation of sigma, SRS(k), has j parts; when no such k exists then T(i, j) = -1.

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