A341969 -id:A341969 - cp's OEIS frontend

A342592 Irregular triangle read by rows in which row n lists the next width pattern not yet encountered as n increases in the symmetric representation of sigma(n), where each contiguous sequence of identical widths w in A249223 is replaced by a single entry of w (see A341969).

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

Offset: 1

Hartmut F. W. Hoft, Mar 16 2021

Every row in the table representing a width pattern w contains an odd number 2*k - 1, k >= 1, of entries where k is the number of odd divisors of (the smallest) number n (first column in the triangle, see also A342594) whose symmetric representation of sigma(n) has width pattern w.
Not every possible Dyck path (see A001405) is realized as the width pattern in the symmetric representation of sigma. The (lexicographically) smallest such pattern w = ( 1 0 1 2 1 2 1 0 1 ) of length 9 would have to be realized by a number n = 2^s * p^4, s >= 0 and p an odd prime, that has 5 odd divisors; however pattern w requires 2^(s+1) < p and 2^(s+1) * p > p^2.
There are only k distinct width patterns of length 2*k - 1 when k is an odd prime. The values in the left parts of the k patterns are: w_i(j) = j, 1 <= j <= i <= k; w_i(j) = i-1, i+1 <= j <= k and j - i odd; w_i(j) = i, i+1 <= j <= k and j - i even. This follows from the fact that when for number n = 2^s * p^(k-1) with s >= 0 and k prime, and the leftmost maximum width in pattern w is m then p^m < 2^(s+1) < p^(m+1) so that the width pattern  m  m-1  m  m-1 ... must continue to the center of w.

			The first 7 rows in the irregular triangle below correspond to rows 1, 3, 6, 9, 15, 18 and 21 of the triangle in A249223 of widths. Collapsing each constant section of a row to a single value in A249223 produces the left half of the corresponding row in the triangle of this sequence.
Every power of 2 has the width pattern (1); every odd prime has the width pattern (1, 0, 1); the triangle in A249223 shows that 12 and 24 have the same width pattern ( 1 2 1 ) as number 6.
The first 21 distinct width patterns. The smallest number whose symmetric representation of sigma has the width pattern of that row is listed in the  first column. All possible symmetric Dyck path patterns of lengths 1, 3, 5 and 7 are in the triangle below; their respective counts are A001405(1,2,3,4) = (1,2,3,6).
    1  1
    3  1 0 1
    6  1 2 1
    9  1 0 1 0 1
   15  1 0 1 2 1 0 1
   18  1 2 1 2 1
   21  1 0 1 0 1 0 1
   30  1 2 1 2 1 2 1
   45  1 0 1 2 1 2 1 2 1 0 1
   60  1 2 3 2 3 2 1
   63  1 0 1 0 1 2 1 0 1 0 1
   72  1 2 3 2 1
   75  1 0 1 2 1 0 1 2 1 0 1
   78  1 2 1 0 1 2 1
   81  1 0 1 0 1 0 1 0 1
   90  1 2 1 2 3 2 3 2 1 2 1
  105  1 0 1 2 1 2 1 0 1 2 1 2 1 0 1
  120  1 2 3 4 3 2 1
  135  1 0 1 2 1 2 1 2 1 2 1 2 1 0 1
  147  1 0 1 0 1 0 1 0 1 0 1
  150  1 2 1 2 1 2 1 2 1 2 1

Subsequence of A341969.

Cf. A001405, A235791, A237048, A237270, A237591, A237593, A249223, A342594.

(* function a341969[ ] is defined in A341969*)
a342592[n_] := Module[{listW={}, k, w}, For[k=1, k<=n, k++, w=a341969[k]; If[!MemberQ[listW, w], AppendTo[listW, w]]]; Flatten[listW]]
a342592[81] (* 15 distinct patterns; the 15th is encountered at n=81 *)

A377667 Square array read by antidiagonals upwards: T(i,j) is the smallest number m such that the symmetric representation of sigma, SRS(m), has maximum width 3, consists of i parts and has 2*j occurrences of maximum width 3 in its width pattern (row m of A341969).

Original entry on oeis.org

60, 10728, 210, 315, 7620, 810, 495, 1155, 840456, 2070, 525, 28158, 945, 88410, 7290, 1275, 1995, 30555, 1575, 408150, 12810, 1287, 2625, 3003, 22365, 2835, 1313010, 45450, 6105, 3315, 10659, 18975, 382305, 11385

Offset: 1

Hartmut F. W. Hoft, Nov 03 2024

When m is odd and SRS(m) has maximum width 3 then SRS(m) has at least 3 parts because the first and last parts of SRS(m) consist of a single leg of width 1. Therefore, the first two rows of the table contain only even numbers. The numbers in the third row appear to be odd and divisible by 15.

			a(8) = T(3,2) = 1155 is the smallest example whose symmetric representation of sigma has 3 parts and 4 counts of width 3 in its width pattern.
Upper left hand section of table T(i, j) = m, numbers m <= 10^7, Columns j indicate 2j occurrences of width 3 in the width pattern of m. T(2, 7) > 10^7.
i\j| 1       2       3       4       5       6       7       8    ...
---------------------------------------------------------------------
1  | 60      210     810     2070    7290    12810   45450   146610
2  | 10728   7620    840456  88410   408150  1313010 >10^7   8596710
3  | 315     1155    945     1575    2835    11385   8505    40095
4  | 495     28158   30555   22365   382305  296835  256095  199395
5  | 525     1995    3003    18975   15147   23925   14553   186219
6  | 1275    2625    10659   35217   132957  818363  312039  1760031
7  | 1287    3315    13125   37107   44289   195415  482937  258687
8  | 6105    3861    31875   65625   132153  149435  807495  1426113
9  | 3591    10773   56889   66861   254065  797979  319599  2199477
10 | 6783    16443   57477   222999  417175  1540875 768339  4670991
11 | 18963   35397   106191  965979  1025973 1770783 2489151 7547427
12 | 90801   58653   47481   1223365 2449785 4600617 ...     ...
13 | 152019  107457  817209  2213253 1740081 4310481
14 | 257397  297087  410571  3086349 3552213 5170055
15 | 335225  815409  1360989 2079609 ...     ...
16 | 1523319 2600283 1642557 2563239
17 | 1473725 1739375 4116777 ...
18 | 4008125 3826625 3687475
19 | 7576085 7937875 ...
...  ...     ...

Subsequence of A376829.

Cf. A237591, A237593, A249223, A341969.

(* widthPattern[ ] and its support functions are defined in A376829 *)
t377667[b_, {r_, c_}] := Module[{t=ConstantArray[0, {r, c}], k, wP, c3, p3}, For[k=1, k<=b, k++, wP=widthPattern[k]; If[Max[wP]==3, c3=Count[wP, 3]; If[EvenQ[c3]&&c3<=2c, c3/=2; p3=Length[Select[SplitBy[wP, #!=0&], First[#]!=0&]]; If[p3<=r &&t[[p3, c3]]==0, t[[p3, c3]]=k]]]]; t]
t377667[1540875, {10, 6}] (* complete 10 x 6 upper left hand section of table *)

A377668 Square array read by antidiagonals upwards: T(i,j), i, j >= 1, is the smallest number m such that the symmetric presentation of sigma, SRS(m), has maximum width 3, consists of 2i-1 parts and has 2j-1 occurrences of maximum width 3 in its width pattern (row m of A341969).

Original entry on oeis.org

72, 2450, 648, 1225, 120050, 450, 3969, 581042, 211250, 20808, 9801, 30625

Offset: 1

Hartmut F. W. Hoft, Nov 03 2024

Maximum width 3 can occur an odd number of times in the width pattern of SRS(m) only for numbers m in this sequence for which SRS(m) has an odd number of parts. In that case width 3 must occur at the diagonal of SRS(m). However, the center part of SRS(m) need not be unimodal.

			For a(1) = 72 SRS(a(1)) is unimodal: 12321.
For a(2) = 2450 the center part of SRS(a(2)) is not unimodal: 1212123212121.
For a(11) = 9801 SRS(a(11)) consists of 9 unimodal parts with maximum width in successive parts nondecreasing to the center part of SRS(a(11)); its width pattern is: 1 0 1 0 1 2 1 0 1 2 1 0 1 2 3 2 1 0 1 2 1  0 1 2 1 0 1 0 1.
Ragged upper left hand section of table T(i, j) = m, numbers m <= 10^7, rows i denoting 2*i-1 parts in SRS(m) and columns j denoting 2*j-1 occurrences of width 3 in the width pattern of SRS(m):
i\j  1       2       3       4       5       6       7    ...
-------------------------------------------------------------
1  | 72      648     450     20808   27378   11250   1996002
2  | 2450    120050  211250  61250   81225   5281250 1531250
3  | 1225    581042  >10^7   354025  >10^7   148225  442225
4  | 3969    30625   321489  127449  1500625 2393209
5  | 9801    6175225 765625  1375929         648025
6  | 4809249 88209   2082249 983961
7  | 385641  1185921 159201  >10^7
8  | 5461569 3470769         7144929
9  | 7177041
10 | 8497225
...

Subsequence of A376829.

Cf. A237591, A237593, A249223, A341969, A377667.

(* widthPattern[ ] and its support functions are defined in A376829 *)
t377668[b_, {r_, c_}] := Module[{t=ConstantArray[0, {r, c}], k, wP, c3, p3}, For[k=1, k<=b, k++, wP=widthPattern[k]; If[Max[wP]==3, c3=Count[wP, 3]; If[OddQ[c3]&&c3+1<=2c, c3=(c3+1)/2; p3=Length[Select[SplitBy[wP, #!=0&], First[#]!=0&]]; If[OddQ[p3]&&p3+1<=2r, p3=(p3+1)/2; If[t[[p3, c3]]==0, t[[p3, c3]]=k]]]]]; t]
t377668[581042, {4, 4}] (* initial 4x4 section except for T(3, 3) > 10^7 *)

A279387 Irregular triangle read by rows: suppose the symmetric representation of sigma(n) consists of m = A250068(n) layers of width 1, arranged in increasing order; then T(n,k) (n >= 1, 1 <= k <= m) is the number of subparts in the k-th layer.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 1, 2, 1, 3, 2, 2, 1, 1, 2, 2, 3, 1, 1, 2, 1, 2, 2, 1, 1, 4, 2, 2, 1, 1, 3, 2, 4, 1, 1, 2, 1, 3, 2, 1, 4, 2, 3, 1, 1, 2, 2, 2, 4, 1, 1, 2, 1, 3, 2, 2, 3, 3, 2, 2, 1, 1, 3, 3, 4, 2, 2, 1, 3, 4, 1, 1, 4, 2, 2, 1, 1, 2, 2, 2, 5, 1, 1, 4, 1, 3, 2, 2, 4, 3, 1, 2, 1, 1, 1, 2, 2

Offset: 1

Omar E. Pol, Dec 12 2016

The "subparts" of the symmetric representation of sigma(n) are defined to be the regions that arise after the dissection of the symmetric representation of sigma(n) into successive layers of width 1.
The number of layers of width 1 in the symmetric representation of sigma(n) is given in A250068.
The number of subparts in the first layer of the symmetric representation of sigma(n) is equal to A237271(n).
We can find the symmetric representation of sigma(n) as the terraces at the n-th level (starting from the top) of the stepped pyramid described in A245092.
(All above comments are essentially the same as the comments dated Nov 05 2016 at the old version of A275601, which was the same as A001227).
The sum of row n equals the number of subparts in the symmetric representation of sigma(n).
Conjecture:
The number of subparts in the symmetric representation of sigma(n) equals A001227(n), the number of odd divisors of n.
From Hartmut F. W. Hoft, Dec 16 2016: (Start)
Proof:
Each row of the irregular triangle of A262045 can be interpreted as a step function of step sizes 1, 0, and -1. The numbers in row n are the widths of the segments in the parts of the symmetric representation of sigma(n). Each new subpart in a segment (in the left half) of row n starts at the same odd index that represents an odd divisor d of n in the irregular triangle of A237048. Either a subpart ends at an even index e, representing a second odd divisor, which satisfies d * e = oddpart(n), and thus the entire subpart is duplicated in the symmetric portion of the representation, or a subpart runs through the center and continues contiguously into the right half of the symmetric portion of the representation. In other words, the number of subparts in row n equals the number of odd divisors of n, i.e., the conjecture is true. (End)

			Triangle begins (first 18 rows):
1;
1;
2;
1;
2;
1, 1;
2;
1;
3;
2;
2;
1, 1;
2;
2;
3, 1;
1;
2;
1, 2;
...
For n = 12, the 11th row of triangle A237593 is [6, 3, 1, 1, 1, 1, 3, 6] and the 12th row of the same triangle is [7, 2, 2, 1, 1, 2, 2, 7], so the diagram of the symmetric representation of sigma(12) = 28 is constructed as shown below in Figure 1:
.                          _                                    _
.                         | |                                  | |
.                         | |                                  | |
.                         | |                                  | |
.                         | |                                  | |
.                         | |                                  | |
.                    _ _ _| |                             _ _ _| |
.                  _|    _ _|                           _|  _ _ _|
.                _|     |                             _|  _| |
.               |      _|                            |  _|  _|
.               |  _ _|                              | |_ _|
.    _ _ _ _ _ _| |    28                 _ _ _ _ _ _| |    5
.   |_ _ _ _ _ _ _|                      |_ _ _ _ _ _ _|
.                                                       23
.
.   Figure 1. The symmetric            Figure 2. After the dissection
.   representation of sigma(12)        of the symmetric representation
.   has only one part which            of sigma(12) into layers of
.   contains 28 cells, so              width 1 we can see two "subparts"
.   A237271(12) = 1.                   that contain 23 and 5 cells
.                                      respectively, so the 12th row of
.                                      this triangle is [1, 1], and the
.                                      row sum is A001227(12) = 2,
.                                      equaling the number of odd divisors
.                                      of 12.
.
For n = 15, the 14th row of triangle A237593 is [8, 3, 1, 2, 2, 1, 3, 8] and the 15th row of the same triangle is [8, 3, 2, 1, 1, 1, 1, 2, 3, 8], so the diagram of the symmetric representation of sigma(15) = 24 is constructed as shown below in Figure 3:
.                                _                                  _
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                          _ _ _|_|                           _ _ _|_|
.                      _ _| |      8                      _ _| |      8
.                     |    _|                            |  _ _|
.                    _|  _|                             _| |_|
.                   |_ _|  8                           |_ _|  1
.                   |                                  |    7
.    _ _ _ _ _ _ _ _|                   _ _ _ _ _ _ _ _|
.   |_ _ _ _ _ _ _ _|                  |_ _ _ _ _ _ _ _|
.                    8                                  8
.
.   Figure 3. The symmetric            Figure 4. After the dissection
.   representation of sigma(15)        of the symmetric representation
.   has three parts of size 8          of sigma(15) into layers of
.   because every part contains        width 1 we can see four "subparts".
.   8 cells, so A237271(15) = 3.       The first layer has three subparts:
.                                      [8, 7, 8]. The second layer has
.                                      only one subpart of size 1, so
.                                      the 15th row of this triangle is
.                                      [3, 1], and the row sum is
.                                      A001227(15) = 4, equaling the
.                                      number of odd divisors of 15.
.
For n = 360, the 359th row of triangle A237593 is [180, 61, 30, 19, 12, 9, 7, 6, 4, 4, 3, 3, 2, 3, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1] and the 360th row of the same triangle is [181, 60, 31, 18, 13, 9, 7, 5, 5, 4, 3, 2, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1], so have that the symmetric representation of sigma(360) = 1170 has only one part, five layers, and six subparts: [(719), (237), (139), (71), (2, 2)], so the 360th row of this triangle is [1, 1, 1, 1, 2], and the row sum is A001227(360) = 6, equaling the number of odd divisors of 360 (the diagram is too large to include).
From _Hartmut F. W. Hoft_, Dec 16 2016: (Start)
45 has 6 subparts of which 2 have symmetric duplicates and 2 span the center. Row length is 18 and "|" indicates the center marker for a row.
1 2 3 4 5 6 7 8 9|9 8 7 6 5 4 3 2 1  : position indices
1 0 1 1 2 1 1 1 2|2 1 1 1 2 1 1 0 1  : row 45 of A262045
1   1 1 1 1 1 1 1|1 1 1 1 1 1 1   1  : layer 1
        1       1|1       1          : layer 2
1 1 1 0 1 1 0 0 1|                   : row 45 of A237048 (odd divisors)
+ - + . + - . . +|                   : change in level ("." no change)
90 has 6 subparts and 3 layers (row length is 24).
1 2 3 4 5 6 7 8..10..12|.14..16..18..20..22..24 : position indices
1 1 2 1 2 2 2 2 3 3 3 2|2 3 3 3 2 2 2 2 1 2 1 1 : row 90 of A262045
1 1 1 1 1 1 1 1 1 1 1 1|1 1 1 1 1 1 1 1 1 1 1 1 : layer 1
    1   1 1 1 1 1 1 1 1|1 1 1 1 1 1 1 1   1     : layer 2
                1 1 1  |  1 1 1                 : layer 3
1 0 1 1 1 0 0 0 1 0 0 1|                        : row 90 of A237048
+ . + - + . . . + . . -|                        : change in level ("." no change)
The process of successive levels provides two "default" dissections of the symmetric representation into subparts from the boundary at n towards the boundary at n-1 or in the reverse direction. (End)
From _Omar E. Pol_, Nov 24 2020: (Start)
For n = 18 we have that the 17th row of triangle A237593 is [9, 4, 2, 1, 1, 1, 1, 2, 4, 9] and the 18th row of the same triangle is [10, 3, 2, 2, 1, 1, 2, 2, 3, 10], so the diagram of the symmetric representation of sigma(18) = 39 is constructed as shown below in Figure 5:
.                                     _                                      _
.                                    | |                                    | |
.                                    | |                                    | |
._                                   | |                                    | |
.                                    | |                                    | |
.                                    | |                                    | |
.                                    | |                                    | |
.                                    | |                                    | |
.                                    | |                                    | |
.                             _ _ _ _| |                             _ _ _ _| |
.                            |    _ _ _|                            |  _ _ _ _|
.                           _|   |                                 _| | |
.                         _|  _ _|                               _|  _|_|
.                     _ _|  _|                               _ _|  _|    2
.                    |     |  39                            |  _ _|
.                    |  _ _|                                | |_ _|
.                    | |                                    | |    2
.   _ _ _ _ _ _ _ _ _| |                   _ _ _ _ _ _ _ _ _| |
.  |_ _ _ _ _ _ _ _ _ _|                  |_ _ _ _ _ _ _ _ _ _|
.                                                              35
.
.   Figure 5. The symmetric               Figure 6. After the dissection
.   representation of sigma(18)           of the symmetric representation
.   has one part of size 39, so           of sigma(18) into layers of
.   A237271(18) = 1.                      width 1 we can see three "subparts".
.                                         The first layer has one subpart of
.                                         size 35. The second layer has
.                                         two subparts of size 2, so
.                                         the 18th row of this triangle is
.                                         [1, 2], and the row sum is
.                                         A001227(18) = 3.
(End)

The sum of row n equals A001227(n).
Hence, if n is odd, the sum of row n equals A000005(n).
Row n has length A250068(n).
Column 1 gives A237271.
For more information about "subparts" see A279388 and A279391.
Cf. A000203, A196020, A235791, A236104, A237048, A237270, A237591, A237593, A239657, A243982, A244050, A245092, A249223, A249351, A250070, A262045, A262611, A261699, A262626, A279693, A280850, A280851, A296508.
Cf. A235791, A237048, A237270, A237591, A237593, A247687, A249223, A250070, A264102, A280851, A341969.

(* function a341969[ ] is defined in A341969 *)
a279387[n_] := Module[{widthL=a341969[n], partL, cL, top, ft, sL}, partL=Select[SplitBy[widthL, #==0&], #!={0}&]; cL=Table[0, Max[widthL]]; While[partL!={}, top=Last[partL]; ft=First[top]; sL=Select[SplitBy[top, #==ft&], #!={ft}&];
cL[[ft]]++; partL=Join[Most[partL], sL]]; cL]
Flatten[a279387[74]] (* the first 74 rows of the table; Hartmut F. W. Hoft, Feb 24 2021 *)

Definition edited by Omar E. Pol and N. J. A. Sloane, Nov 25 2020

A239663 a(n) is the smallest number k such that the symmetric representation of sigma(k) has n parts.

Original entry on oeis.org

1, 3, 9, 21, 63, 147, 357, 903, 2499, 6069, 13915, 29095, 59455, 142945, 320045, 643885, 1367465, 3287735, 6779135, 13853015, 30262595, 61773745

Offset: 1

Omar E. Pol, Mar 23 2014

Conjecture 1: where records occur in A237271. - Omar E. Pol, Dec 27 2016
For more information about the symmetric representation of sigma see A237270, A237593.
This sequence of (first occurrence of) parts appears to be strictly increasing in contrast to sequence A250070 of (first occurrence of) maximum widths. - Hartmut F. W. Hoft, Dec 09 2014
Conjecture 2: all terms are odd numbers. - Omar E. Pol, Oct 14 2018
Proof of Conjecture 2: Let n = 2^m * q with m>0 and q odd; then the 1's in even positions of row n in the triangle of A237048 are at positions 2^(m+1) * d <= row(n) where d divides q. For n/2 the even positions of 1's occur at the smaller values 2^m * d <= row(n/2), thus either keeping or reducing widths (A249223) of parts in the symmetric representation of sigma for n/2 inherited from row n. Therefore the number of parts for n is at most as large as for n/2, i.e., all numbers in this sequence are odd. - Hartmut F. W. Hoft, Sep 22 2021
Observation: at least for n = 1..21 we have that 2*a(n) < a(n+1). - Omar E. Pol, Sep 22 2021
From Omar E. Pol, Jul 28 2025: (Start)
Conjecture 3: a(n) is the smallest number k having n 2-dense sublists of divisors of k.
The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.
In a sublist of divisors of k the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of k.
An example of the conjecture 3 for n = 1..5 is as shown below:
  ----------------------------------------------------
  |    |  List of divisors of k         |     |      |
  |  k |  [with sublists in brackets]   |  n  | a(n) |
  ----------------------------------------------------
  |  1 |  [1];                          |  1  |   1  |
  |  3 |  [1], [3];                     |  2  |   3  |
  |  9 |  [1], [3], [9];                |  3  |   9  |
  | 21 |  [1], [3], [7], [21];          |  4  |  21  |
  | 63 |  [1], [3], [7, 9], [21], [63]; |  5  |  63  |
(End)
Conjecture 4: a(n) is the smallest number k having n divisors p of k such that p is greater than twice the adjacent previous divisor of k. - Omar E. Pol, Aug 05 2025

			------------------------------------------------------
n       a(n)     A239665                  A266094(n)
------------------------------------------------------
1        1       [1]                           1
2        3       [2, 2]                        4
3        9       [5, 3, 5]                    13
4       21       [11, 5, 5, 11]               32
5       63       [32, 12, 16, 12, 32]        104
...
For n = 3 the symmetric representation of sigma(9) = 13 contains three parts [5, 3, 5] as shown below:
.
.     _ _ _ _ _ 5
.    |_ _ _ _ _|
.              |_ _ 3
.              |_  |
.                |_|_ _ 5
.                    | |
.                    | |
.                    | |
.                    | |
.                    |_|
.

Hartmut F. W. Hoft, Procedural implementation for extension values
Omar E. Pol, Perspective view of the stepped pyramid (16 levels)

Row 1 of A240062.
Cf. A000203, A196020, A236104, A235791, A237048, A237270, A237271, A237591, A237593, A238443, A239657, A239660, A239665, A239931-A239934, A245092, A262626, A266094.
Cf. A249223, A250070, A262045, A320521, A341969, A341970, A341971, A347980.
Cf. A174973 (2-dense numbers), A384149, A384222, A384225, A384226.

(* a239663[] permits computation in intervals *)
(* Function a237270[] is defined in A237270 *)
(* variable "list" contains the first occurrences up to m *)
a239663[list_,{m_, n_}]:=Module[{firsts=list, g=Length[list], i, p}, For[i=m, i<=n, i++, p=Length[a237270[i]]; If[p>g, AppendTo[firsts, i]; g=p]]; firsts]
a239663[{1}, {1, 1000}] (* computes the first 8 values *)
(* Hartmut F. W. Hoft, Jul 08 2014 *)
(* support functions are defined in A341969, A341970 & A341971 *)
a239663[n_, len_] := Module[{list=Table[0, len], i, v}, For[i=1, i<=n, i+=2, v=Count[a341969[i], 0]+1;If[list[[v]]==0, list[[v]]=i]]; list]
a239663[62000000,22] (* Hartmut F. W. Hoft, Sep 22 2021 *)

a(6)-a(8) from Michel Marcus, Mar 28 2014
a(9) from Michel Marcus, Mar 29 2014
a(10)-a(11) from Michel Marcus, Apr 02 2014
a(12) from Hartmut F. W. Hoft, Jul 08 2014
a(13)-a(18) from Hartmut F. W. Hoft, Dec 09 2014
a(19)-a(22) from Hartmut F. W. Hoft, Sep 22 2021

A240062 Square array read by antidiagonals in which T(n,k) is the n-th number j with the property that the symmetric representation of sigma(j) has k parts, with j >= 1, n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 6, 7, 15, 21, 8, 10, 25, 27, 63, 12, 11, 35, 33, 81, 147, 16, 13, 45, 39, 99, 171, 357, 18, 14, 49, 51, 117, 189, 399, 903, 20, 17, 50, 55, 153, 207, 441, 987, 2499, 24, 19, 70, 57, 165, 243, 483, 1029, 2709, 6069, 28, 22, 77, 65, 195, 261, 513, 1113

Offset: 1

Omar E. Pol, Apr 06 2014

This is a permutation of the positive integers.
All odd primes are in column 2 (together with some even composite numbers) because the symmetric representation of sigma(prime(i)) is [m, m], where m = (1 + prime(i))/2, for i >= 2.
The union of all odd-indexed columns gives A071562, the positive integers that have middle divisors. The union of all even-indexed columns gives A071561, the positive integers without middle divisors. - Omar E. Pol, Oct 01 2018
Each column in the table of A357581 is a subsequence of the respective column in the table of this sequence; however, the first row in the table of A357581 is not a subsequence of the first row in the table of this sequence. - Hartmut F. W. Hoft, Oct 04 2022
Conjecture: T(n,k) is the n-th positive integer with k 2-dense sublists of divisors. - Omar E. Pol, Aug 25 2025

			Array begins:
   1,  3,  9, 21,  63, 147, 357,  903, 2499, 6069, ...
   2,  5, 15, 27,  81, 171, 399,  987, 2709, 6321, ...
   4,  7, 25, 33,  99, 189, 441, 1029, 2793, 6325, ...
   6, 10, 35, 39, 117, 207, 483, 1113, 2961, 6783, ...
   8, 11, 45, 51, 153, 243, 513, 1197, 3025, 6875, ...
  12, 13, 49, 55, 165, 261, 567, 1239, 3087, 6909, ...
  16, 14, 50, 57, 195, 275, 609, 1265, 3249, 7011, ...
  18, 17, 70, 65, 231, 279, 621, 1281, 3339, 7203, ...
  20, 19, 77, 69, 255, 297, 651, 1375, 3381, 7353, ...
  24, 22, 91, 75, 273, 333, 729, 1407, 3591, 7581, ...
  ...
[Lower right hand triangle of array completed by _Hartmut F. W. Hoft_, Oct 04 2022]

Index entries for sequences that are permutations of the natural numbers

For programs see A237271 and A237593.
Row 1 is A239663.
Column 1 is A174973.
Columns 2..7: A239929, A279102, A280107, A320066, A320511, A357775.
For more information see A239663 and A239665.
Cf. A000203, A006254, A065091, A067742, A071561, A071562, A196020, A236104, A235791, A237048, A237270, A239660, A239929, A239931-A239934, A245092, A262626, A319529, A319796, A319801, A319802.
Cf. A341969, A341970, A341971, A357581.

(* function a341969 and support functions are defined in A341969, A341970 and A341971 *)
partsSRS[n_] := Length[Select[SplitBy[a341969[n], #!=0&], #[[1]]!=0&]]
widthTable[n_, {r_, c_}] := Module[{k, list=Table[{}, c], parts}, For[k=1, k<=n, k++, parts=partsSRS[k]; If[parts<=c&&Length[list[[parts]]]=1, j--, vec[[PolygonalNumber[i+j-2]+j]]=arr[[i, j]]]]; vec]
a240062T[n_, r_] := TableForm[widthTable[n, {r, r}]]
a240062[6069, 10] (* data *)
a240062T[7581, 10] (* 10 X 10 array - Hartmut F. W. Hoft, Oct 04 2022 *)

a(n) > 128 from Michel Marcus, Apr 08 2014

A342594 Earliest occurrence of the next distinct width pattern (as listed in A342592) in the symmetric representation of sigma(n) not yet encountered as n increases.

Original entry on oeis.org

1, 3, 6, 9, 15, 18, 21, 30, 45, 60, 63, 72, 75, 78, 81, 90, 105, 120, 135, 147, 150, 162, 165, 180, 189, 210, 225, 231, 300, 315, 357, 360, 378, 390, 405, 420, 441, 450, 465, 495, 504, 525, 540, 567, 630, 648, 666, 675, 690, 693, 729, 735, 770, 810, 825, 840, 855, 858, 882, 900, 903, 945, 975, 990

Offset: 1

Hartmut F. W. Hoft, Mar 16 2021

nonn

The width pattern of the symmetric representation of sigma(a(n)) is the n-th row of the table of A342592.

Conjecture: If for some number n the symmetric representation of sigma(n) has the symmetric width pattern w in row n of A342592 then infinitely many numbers have that width pattern w.

			a(1) = 1 is the smallest power of 2 whose symmetric representation of sigma has width pattern (1).
a(2) = 3 is the smallest odd prime whose symmetric representation of sigma has width pattern (1 0 1).
a(4) = 9 is the first number whose symmetric representation of sigma has width pattern (1 0 1 0 1). The infinitely many numbers 2^s * p^2, s >= 0 and p an odd prime larger than 2^(s+1), have the same width pattern.

Cf. A235791, A237048, A237270, A237591, A237593, A249223, A341969, A342592.

(* function a341969[ ] is defined in A341969 *)
a342594[n_] := Module[{listW={}, listK={}, k, w}, For[k=1, k<=n, k++, w=a341969[k]; If[!MemberQ[listW, w], AppendTo[listW, w]; AppendTo[listK, k]]]; listK]
a342594[990] (* 64 entries; the 64th new pattern is encountered at n=990 *)

A318843 a(n) is the smallest number k such that the symmetric representation of sigma(k) consists of n parts of width 1.

Original entry on oeis.org

1, 3, 9, 21, 81, 147, 729, 903, 3025, 6875, 59049, 29095, 531441, 171875, 366025, 643885, 43046721, 3511475

Offset: 1

Hartmut F. W. Hoft, Sep 04 2018

The sequence is infinite since, for example, for any n >= 1 the symmetric representation of sigma(3^n) consists of n + 1 parts of width 1. However, it is not increasing since a(11) = 59049 = 3^10 and a(12) = 29095 = 5 * 11 * 23^2. Also a(13) <= 531441 = 3^12.
This sequence is a subsequence of A174905; its subsequences a(n) for odd/even n are subsequences of A241010/A241008, respectively. Some even-indexed elements of this sequence are members of A239663, e.g., a(2), a(4), a(6), a(8) and a(12), but not a(10) = 6875.
The central pair of parts in the symmetric representation of sigma(a(2)), sigma(a(4)) and sigma(a(8)) meets at the diagonal (see A298856).
From Hartmut F. W. Hoft, Oct 04 2021: (Start)
An upper bound to the sequence is a(n) <= 3^(n-1), n >= 1, (see A348171).
For p = 1,2,3,5,7,11,13,17, a(p) = 3^(p-1) and this equality possibly holds for all a(p) with p a prime.
Also, 75 * 10^6 < a(19) <= 3^18, a(20) = 15391255, a(21) = 44289025 and a(n) > 75 * 10^6 for n > 21.
a(13)-a(18) computations based on A348171 rather than A237270.
The symmetric representation of sigma(3^(p-1)), p prime, consists of p parts and its middle part has area 3^((p-1)/2). (End)
a(n) >= A038547(n) with equality for n=1 and primes n since the distinct prime divisors of n can be replaced by primes 3, 5, 7, 11, ... yielding a smaller number k with the same number of odd divisors. However, some parts in the symmetric representation of sigma(k) have width at least 2. - Hartmut F. W. Hoft, Dec 11 2023

			The smallest number k whose symmetric representation of sigma(k) consists of four parts of width one is a(4) = 21. The parts are 11, 5, 5, 11.
a(4) = 3*7 has width pattern, A341969, 1010101 while A038547(4) = 3*5 has width pattern 1012101. a(6) = 3 * 7^2 = 147 has width pattern 10101010101 while A038547(6) = 3^2 * 5 = 45 has width pattern 10121212101. - _Hartmut F. W. Hoft_, Dec 11 2023

Cf. A174905, A237048, A237270, A237271, A237593, A239663, A239665, A240062, A240542, A241008, A241010, A249351, A298856.
Cf. A249223, A348171.
Cf. A038547, A341969.

(* Function path[] is defined in A237270 *)
segmentsSR[pathN0_, pathN1_] := SplitBy[Map[Min, Drop[Drop[pathN0, 1], -1] - pathN1], #==0&]
regions[pathN0_ ,pathN1_] := Select[Map[Apply[Plus, #]&, segmentsSR[pathN0, pathN1]], #!=0&]
width1Q[pathN0_, pathN1_] := SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[pathN0, 1], -1] - pathN1, 1]]]
(* parameter seq is the list of elements of the sequence in interval 1..m-1 already computed with an entry of 0 representing an element not yet found *)
a318843[m_, n_, seq_] := Module[{list=Join[seq, Table[0, 10]], path1=path[m-1], path0, k, a, r, w}, For[k=m, k<=n, k++, path0=path[k]; a=regions[path0, path1]; r=Length[a]; w=width1Q[path0, path1]; If[w && list[[r]]==0, list[[r]]=k]; path1=path0]; list]
a318843[2,60000,{1}] (* data - actually computed in steps *)

a(13)-a(18) from Hartmut F. W. Hoft, Oct 04 2021

A279029 Numbers k with the property that the smallest and the largest Dyck path of the symmetric representation of sigma(k) do not share line segments.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 136, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252, 256

Offset: 1

Omar E. Pol, Dec 08 2016

nonn

Numbers k such that the symmetric representation of sigma(k) is formed by only one part, or that it's formed by only two parts and they meet at the center.
Numbers k whose total length of all line segments of the symmetric representation of sigma(k) is equal to 4*k (cf. A348705). For the positive integers k that are not in this sequence the mentioned total length is < 4*k. - Omar E. Pol, Nov 02 2021
From Hartmut F. W. Hoft, Jan 25 2025: (Start)
The following three statements are equivalent for numbers k >= 1:
(1) The symmetric representation of sigma(k) is formed of 2 parts that meet at the diagonal.
(2) A249223(k, A003056(k)) = 0 is the only 0 in row k of the triangle, and A237591(k, A003056(k)) = 1.
(3) Row k of the triangle in A341969 contains a single 0 at the center position.
The following two statements are equivalent for numbers k >= 1:
(1) The symmetric representation of sigma(k) consists of a single part.
(2) Row k of the triangle in A249223 contains no 0. (End)
This sequence is the disjoint union of A262259 and A174973. Each member of A262259 has the form k = q*(2*q + 1) where 2*q + 1 is prime; also A003056(k) = 2*q. Therefore [q, 2*q] contains a divisor q of k while (q, 2*q] contains no divisor of k. A262259 is a subsequence of A298259, see also A240542. - Hartmut F. W. Hoft, Mar 24 2025
My two links below give detailed proofs for the last comment.  - Hartmut F. W. Hoft, Jun 10 2025

			1, 2, 3, 4, 6, 8, 10, 12 and 16 are in the sequence because the smallest and the largest Dyck path of their symmetric representation of sigma do not share line segments, as shown below.
Illustration of initial terms:
  n
  .      _ _ _ _   _   _   _   _       _
  1     |_| | | | | | | | | | | |     | |
  2     |_ _|_| | | | | | | | | |     | |
  3     |_ _|  _|_| | | | | | | |     | |
  4     |_ _ _|    _|_| | | | | |     | |
         _ _ _|  _|  _ _|_| | | |     | |
  6     |_ _ _ _|  _| |  _ _|_| |     | |
         _ _ _ _| |_ _|_|    _ _|     | |
  8     |_ _ _ _ _|  _|     |    _ _ _| |
         _ _ _ _ _| |      _|   |  _ _ _|
  10    |_ _ _ _ _ _|  _ _|    _| |
         _ _ _ _ _ _| |      _|  _|
  12    |_ _ _ _ _ _ _|  _ _|  _|
                        |  _ _|
                        | |
         _ _ _ _ _ _ _ _| |
  16    |_ _ _ _ _ _ _ _ _|
  ...

Antti Karttunen, Table of n, a(n) for n = 1..20000
Hartmut F. W. Hoft, Proof: Dyck Paths of SRS(n) meeting only at diagonal
Hartmut F. W. Hoft, Proofs for A279029 = A174973 union A262259

UNION of A174973 and A262259.
Positions of 0's in A279228.
Complement is A279244.
Cf. A000203, A003056, A008586, A174973, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A249223, A262259, A323648, A348705, A379968 (characteristic function).
Cf. A240542, A298856.

(* Function path[] is defined in A237270 *)
a279029Q[n_] := Length[Select[Transpose[{Take[path[n], {2,-2}], path[n-1]}], #[[1]]==#[[2]]&]]<=1
a279029[n_] := Select[Range[n], a279029Q]
a279029[256]
(* Alternate, faster function based on A249223 *)
a003056[n_] := Floor[(Sqrt[8n+1]-1)/2]
t249223[n_] :=FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, a003056[n]]]
a262259Q[n_] := Position[t249223[n], 0]=={{a003056[n]}}&&Last[t237591[n]]==1
a174973Q[n_] := !MemberQ[t249223[n], 0]
a279029[n_] := Select[Range[n], a262259Q[#]||a174973Q[#]&]
a279029[256] (* Hartmut F. W. Hoft, Jan 25 2025 *)

is_A279029 = A379968; \\ Antti Karttunen, Jan 12 2025

a(n) = A323648(n-1) + 1, for n >= 2. - Hartmut F. W. Hoft, Jan 25 2025

A342595 Irregular triangle of A342592 read by rows arranged first by length of the width pattern and then lexicographically within blocks of patterns of equal length.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 1, 2, 3, 2, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 2, 1, 0, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 2, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 2, 3, 2, 3, 2, 1

Offset: 1

Hartmut F. W. Hoft, Mar 16 2021

Let u and v be two symmetric width patterns; then u < v if u is shorter than v, and if they have the same length then they are ordered lexicographically, i.e., if i is the first index where u and v differ and u(i) < v(i) then u < v.
This sequence is a permutation of the rows of the irregular triangle in A342592. Every row in the triangle representing a width pattern w contains an odd number 2*k - 1, k >= 1, of entries where k is the number of odd divisors of the smallest number whose symmetric representation of sigma realizes that pattern.
The number of distinct width patterns w of the same length 2*k-1 created by numbers with k odd divisors is computationally challenging since the numbers of their first occurrence can be very large (A342592, A342596). The counts in the table below are already established for n <= 5*10^5 and have not changed through 10^7; the counts are not stable at that larger level for width patterns of numbers with more than 8 odd divisors:
# odd divisors   1   2   3   4    5    6    7    8
pattern count    1   2   3   6    5   16    7   40
A001405          1   2   3   6   10   20   35   70
For any odd number q with k divisors and 2^s < q < 2^(s+1), s >= 0, any number 2^t * q with t > s has the lexicographically largest symmetric width pattern 1 2 3 ... k-2 k-1 k k-1 k-2 ... 3 2 1 of length 2*k - 1. Therefore, the sequence q, 2 * q, 2^2 * q, ... , 2^s * q instantiates at most s+1 different symmetric width patterns; these range from 2 for prime numbers q, patterns (1 0 1) and (1 2 1), to the maximum of s+1 different patterns such as for q = 105 = 3*5*7.

			The number of entries through the center in a row of the triangle below equals the number of odd divisors of any number that has that pattern of widths.
The pattern in row 10 of the triangle below, realized first by n = 30 which labels the row is the smallest number with width pattern (1 2 1 2 1 2 1); 42, 54 and 66 are the other numbers less than 100 realizing that pattern.
The triangle below lists the first 21 distinct symmetric width patterns in the order described above. The smallest number whose symmetric representation of sigma has the width pattern of that row is listed as first column (see A342596). All possible symmetric width patterns of lengths 1, 3, 5 and 7 are realized in the triangle below; their respective counts are A001405(1,2,3,4) = (1,2,3,6).
     1   1
     3   1 0 1
     6   1 2 1
     9   1 0 1 0 1
    18   1 2 1 2 1
    72   1 2 3 2 1
    21   1 0 1 0 1 0 1
    15   1 0 1 2 1 0 1
    78   1 2 1 0 1 2 1
    30   1 2 1 2 1 2 1
    60   1 2 3 2 3 2 1
   120   1 2 3 4 3 2 1
    81   1 0 1 0 1 0 1 0 1
   162   1 2 1 2 1 2 1 2 1
   648   1 2 3 2 3 2 3 2 1
  1296   1 2 3 4 3 4 3 2 1
  5184   1 2 3 4 5 4 3 2 1
   147   1 0 1 0 1 0 1 0 1 0 1
    63   1 0 1 0 1 2 1 0 1 0 1
    75   1 0 1 2 1 0 1 2 1 0 1
    45   1 0 1 2 1 2 1 2 1 0 1

Cf. A001405, A237593, A249223, A341969, A342592, A342594, A342596.

(* function a341969[ ] is defined in A341969 *)
lexicographic[s1_, s2_] := Module[{k=1}, While[s1[[2, k]]==s2[[2, k]], k++]; s1[[2,k]]

cp's OEIS Frontend

A342592 Irregular triangle read by rows in which row n lists the next width pattern not yet encountered as n increases in the symmetric representation of sigma(n), where each contiguous sequence of identical widths w in A249223 is replaced by a single entry of w (see A341969).

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A377667 Square array read by antidiagonals upwards: T(i,j) is the smallest number m such that the symmetric representation of sigma, SRS(m), has maximum width 3, consists of i parts and has 2*j occurrences of maximum width 3 in its width pattern (row m of A341969).

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A377668 Square array read by antidiagonals upwards: T(i,j), i, j >= 1, is the smallest number m such that the symmetric presentation of sigma, SRS(m), has maximum width 3, consists of 2*i-1 parts and has 2*j-1 occurrences of maximum width 3 in its width pattern (row m of A341969).

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A279387 Irregular triangle read by rows: suppose the symmetric representation of sigma(n) consists of m = A250068(n) layers of width 1, arranged in increasing order; then T(n,k) (n >= 1, 1 <= k <= m) is the number of subparts in the k-th layer.

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A239663 a(n) is the smallest number k such that the symmetric representation of sigma(k) has n parts.

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A240062 Square array read by antidiagonals in which T(n,k) is the n-th number j with the property that the symmetric representation of sigma(j) has k parts, with j >= 1, n >= 1, k >= 1.

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A342594 Earliest occurrence of the next distinct width pattern (as listed in A342592) in the symmetric representation of sigma(n) not yet encountered as n increases.

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A318843 a(n) is the smallest number k such that the symmetric representation of sigma(k) consists of n parts of width 1.

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A279029 Numbers k with the property that the smallest and the largest Dyck path of the symmetric representation of sigma(k) do not share line segments.

A377668 Square array read by antidiagonals upwards: T(i,j), i, j >= 1, is the smallest number m such that the symmetric presentation of sigma, SRS(m), has maximum width 3, consists of 2i-1 parts and has 2j-1 occurrences of maximum width 3 in its width pattern (row m of A341969).