A264102 -id:A264102 - cp's OEIS frontend

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.

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

A341969 Irregular triangle read by rows in which row n lists the sequence of widths, each contiguous sequence of identical widths w in A249223 replaced by a single entry of w, in the symmetric representation of sigma(n).

Original entry on oeis.org

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

Offset: 1

Hartmut F. W. Hoft, Feb 24 2021

This sequence is a companion to A279387 in which each contiguous sequence of identical widths w in A249223 are replaced by a single entry of w. Using the resulting distribution pattern of widths across all parts of the symmetric representation of sigma(n) the subparts at each level are counted in A279387.
The sequence of widths are computed first to the diagonal of the symmetric representation of sigma only for those numbers in set F defined in A341971. Then the reversed list less its first number is appended so that the width at the diagonal is not listed twice. Thus every row contains an odd number of entries and is symmetric about its center entry.
Let 1 <= n, 1 <= d <= s = A001227(n) and 1 <= k <= r = floor((sqrt(8*n + 1) - 1)/2). Let Q(n,d) be row n in the triangle of A341970, R(n,d) be row n in the triangle of A341970 and S(n,d) = R(n,Q(n,d)), then T(n,e) = S(n,e) for 1 <= e <= s and T(n,e) = S(n,2*s - e) for s < e <= 2*s - 1 is row n for this sequence.

			The irregular triangle for A279387 and this sequence:
  row  A279387  A341969
  1    1        1
  2    1        1
  3    2        1  0  1
  4    1        1
  5    2        1  0  1
  6    1  1     1  2  1
  7    2        1  0  1
  8    1        1
  9    3        1  0  1  0  1
  10   2        1  0  1
  11   2        1  0  1
  12   1  1     1  2  1
  13   2        1  0  1
  14   2        1  0  1
  15   3  1     1  0  1  2  1  0  1
  16   1        1
  17   2        1  0  1
  18   1  2     1  2  1  2  1
  19   2        1  0  1
  20   1  1     1  2  1
  21   4        1  0  1  0  1  0  1
  ..   ..       ..
  30   1  3     1  2  1  2  1  2  1
  ..   ..       ..
  45   3  3     1  0  1  2  1  2  1  2  1  0  1
  ..   ..       ..
a(17)..a(21) = { 1, 0, 1, 0, 1 } is row 9; the symmetric representation of sigma(9) consists of 3 parts of width 1 - see A247687.
a(37)..a(43) = { 1, 0, 1, 2, 1, 0, 1} is row 15; the symmetric representation of sigma(15) consists of 2 outer parts of width 1 and a central part of width 2 only at the diagonal - see A338488.
a(59)..a(65) = { 1, 0, 1, 0, 1, 0, 1 } is row 21; the symmetric representation of sigma(21) consists of 4 parts of width 1, and 21 is the smallest such number - see A264102.
a(234)..a(240) = { 1, 2, 3, 2, 3, 2, 1 } is row 60; the symmetric representation of sigma(60) consists of 1 part of maximum width 3 which occurs in two subparts, and 60 is the smallest number with width 3 - see A250070.

Cf. A235791, A237048, A237270, A237591, A237593, A247687, A249223, A249351, A250070, A264102, A279387, A280851, A338488, A341970, A341971.

(* function widthL[ ] is defined in A341971 *)
a341969[n_] := Module[{wL=widthL[n]}, Join[wL, Rest[Reverse[wL]]]]
Flatten[Table[a341969[n], {n, 28}]] (* the first 28 rows of the table *)

a(2*A060831(n-1) - (n-1) + e) = T(n,e), 1 <= n, 1 <= e <= 2*A001227(n) - 1.

A280107 Numbers m with the property that the symmetric representation of sigma(m) has four parts.

Original entry on oeis.org

21, 27, 33, 39, 51, 55, 57, 65, 69, 75, 85, 87, 93, 95, 105, 111, 115, 119, 123, 125, 129, 133, 141, 145, 155, 159, 161, 175, 177, 183, 185, 201, 203, 205, 213, 215, 217, 219, 230, 235, 237, 245, 249, 250, 253, 259, 265, 267, 287, 290, 291, 295, 301, 303, 305, 309, 310, 319, 321, 327, 329

Offset: 1

Omar E. Pol, Dec 27 2016

nonn

From Hartmut F. W. Hoft, Jan 27 2018: (Start)
Let n = 2^k * t where k >= 0 and t is odd, and let D be the set of divisors of t less than r(n) = floor((sqrt(8*n+1) - 1)/2). The following statements are equivalent:
(1)  There is exactly one d in D such that 2^(k+1) * d < e where e in D is the next odd divisor larger than d, and the largest divisor f in D satisfies 2^(k+1) * f <= r(n).
(2)  The symmetric representation of sigma(n) consists of four parts.
The property says that the first part of the symmetric representation of n consists of the first 2^(k+1) * d - 1 legs and that the second part starts with leg e and ends with leg 2^(k+1) * f - 1 before or at the middle of the Dyck path (see A237048 and A249223) on the diagonal. Together with their symmetric pair they form the four parts. (End)

			a(1) = 21 because it is the smallest number n whose symmetric representation of sigma(n) has four parts. Note that the sum of the parts is 11 + 5 + 5 + 11 = 32, equaling the sum of the divisors of 21: aigma(21) = 1 + 3 + 7 + 21 = 32.
From _Hartmut F. W. Hoft_, Jan 27 2018: (Start)
230 = 2*5*23 is the first even number since 2^2 < 5, 2^2 * 5 < 23, and row 230 in A237048 has 20 entries with 1's in positions 1, 4, 5, and 20.
Prime number 3 can be a factor for an even number in this sequence as 12246=2*3*13*157 demonstrates with the four parts 12252, 1020, 1020, and 12252 in the symmetric representation of sigma(12246) defined by 1's in positions 1, 3, 4, 12, 13, 39, 52, 156 in row 12246 of A237048; each of the four parts has maximum width 2 and the two central parts meet on the diagonal at 8492. (End)

Column 4 of A240062.
First differs from A264102 at a(10).
Numbers n such that the symmetric representation of sigma(n) has k parts (k = 1..4): A174973 = A238443, A239929, A279102, this sequence.
Cf. A237270, A237271 (number of parts), A237591, A237593, A239665, A245092, A262626.
Cf. A237048, A249223. - Hartmut F. W. Hoft, Jan 27 2018

(* Function a237270[] is defined in A237270 *)
a280107[m_, n_] := Select[Range[m, n], Length[a237270[#]]==4&]
a280107[1, 329] (* data *)
(* Implementation of the property in the Comment section *)
evenPart[n_] := Module[{f=First[FactorInteger[n]]}, If[First[f]!=2, 1, First[f]^Last[f]]]
fourPartsQ[n_] := Module[{e=evenPart[n], oddPart, r=(Sqrt[8*n + 1] - 1)/2, dL}, oddPart=n/evenPart[n]; dL=Select[Divisors[oddPart], #1, 2*e*Last[dL]<=r && Length[Select[2*e*Most[dL]-Rest[dL], #<0&]]==1, False]];
Select[Range[329], fourPartsQ] (* data *)
(* Hartmut F. W. Hoft, Jan 27 2018 *)

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 *)

A365081 Numbers k with the property that the symmetric representation of sigma(k) has four parts and its second part is an octagon of width 1 and one of the vertices of the octagon is also the central vertex of the first valley of the largest Dyck path of the diagram.

Original entry on oeis.org

21, 27, 33, 39, 51, 57, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 201, 213, 219, 237, 249, 267, 291, 303, 309, 321, 327, 339, 381

Offset: 1

Omar E. Pol, Aug 20 2023

Also the row numbers of the triangle A364639 where the rows are [0, 0, 1, 0, -1, 1] or where the rows start with [0, 0, 1, 0, -1, 1] and the remaining terms are zeros.
Observation: the first 29 terms coincide with the first 29 terms of A161345 that are >= 21.
Apparently a(n)=A127329(n) for n>2. - R. J. Mathar, Sep 05 2023

			The symmetric representation of sigma(21) in the first quadrant looks like this:
   _ _ _ _ _ _ _ _ _ _ _
  |_ _ _ _ _ _ _ _ _ _ _|
                        |
                        |
                        |_ _ _
                        |_ _  |_
                            |_ _|_
                                | |_
                                |_  |
                                  | |
                                  |_|_ _ _ _
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          | |
                                          |_|
.
There are four parts (or polygons) and its second part is an octagon of width 1 and one of the vertices of the octagon is also the central vertex of the first valley of the largest Dyck path of the structure so 21 is in the sequence.

Subsequence of A016945, A264102, A280107 and A364414.

Cf. A033676, A161345, A196020, A235791, A236104, A237270 (parts), A237271, A237591, A237593, A240062, A245092, A249351 (widths), A262626, A364639.

A264104 Numbers n with the property that the symmetric representation of sigma(n) has four parts, each of width one and two regions meet at the center of the Dyck path.

Original entry on oeis.org

21, 55, 253, 406, 1081, 1378, 1711, 3403, 3916, 5671, 9316, 11026, 13861, 14878, 15931, 25651, 27028, 34453, 36046, 42778, 50086, 60031, 64261, 73153, 75466, 108811, 114481, 126253, 129286, 154846, 158203, 161596, 171991, 175528, 212878, 258121, 298378, 317206, 326836, 351541, 366796, 371953, 392941

Offset: 1

Hartmut F. W. Hoft, Nov 03 2015

This sequence is a subsequence of A264102 and also of A014105, the second hexagonal numbers. Every number in this sequence is a triangular number.
The sequence A156592 of products of a Sophie Germain prime (A005384) and its associated safe prime (A005385) except for the first pair (2, 5) forms a subsequence of this sequence, the first column in the irregular triangular grid in the example.
The areas of the first two regions are (2^(m+1) - 1) * (2^(m+1) * p^2 * p + 1) / 2 and (2^(m+1) - 1) * (2^(m+1) * p + p + 1) / 2, respectively. Twice their sum equals sigma(n) = (2^(m+1) - 1) * (p + 1) * (2^(m+1) * p + 2).
For a proof of the formula for this sequence see the link.

			406 = 2*7*29 is in the sequence since m = 1 and 4 < 7 < 28 < 29. The first two regions in the symmetric representation of sigma(406) = 720 start with legs 1 and 7 and have areas 306 and 54, respectively. Note also that 406 is a triangular number and the middle two regions meet at the center of the Dyck path.
10 does not belong to this sequence since the symmetric representation of sigma(10) has two regions of width 1 that meet at the diagonal.
There is a natural arrangement of the numbers n = 2^m * p * (2^(m+1) * p + 1) as a sparse irregular triangular (p,m)-grid.
p\m| 0      1       2        3        4        5   ...
-------------------------------------------------------
3  | 21
5  | 55
7  |        406
11 | 253            3916
13 |        1378
17 |                9316
19 |
23 | 1081
29 | 1711           27028
31 |
37 |        11026           175528
41 | 3403
43 |        14878
47 |
53 | 5671                           1439056
59 |                                1783216
61 |                        476776
67 |        36046                            9195616
71 |                161596          2582128
73 |        42778                            10916128
...
The first number in the m = 6 column is 181880128 = 2^6*149*19073 in row p = 149 and the second is 228477376 = 2^6*167*21377 in row p = 167.

Hartmut F. W. Hoft, Diagram of symmetric representations of sigma(n), for n = 21, 55, 253, 406
Hartmut F. W. Hoft, Proof of 4 regions width 1 and 2 meet at center

Cf. A005384, A005385, A014105, A156592, A264102.

For symmetric representation of sigma: A235791, A236104, A237270, A237271, A237591, A237593, A241008, A246955.

mStalk[m_, bound_] := Module[{p=NextPrime[2^(m+1)], list={}}, While[2^m*p*(2^(m+1)*p+1)<=bound, If[PrimeQ[2^(m+1)*p+1], AppendTo[list, 2^m *p*(2^(m+1)*p+1)]]; p=NextPrime[p]]; list]
a264104[bound_] := Module[{m=0, list={}}, While[2^m*NextPrime[2^(m+1)]*(2^(m+1)*NextPrime[2^(m+1)]+1)<=bound, list=Union[list, mStalk[m, bound]]; m++]; list]
a264104[400000] (* data *)

n = 2^m * p * (2^(m+1) * p + 1) where m >= 0, 2^(m+1) < p and p as well as 2^(m+1) * p + 1 are prime.

cp's OEIS Frontend

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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A341969 Irregular triangle read by rows in which row n lists the sequence of widths, each contiguous sequence of identical widths w in A249223 replaced by a single entry of w, in the symmetric representation of sigma(n).

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A280107 Numbers m with the property that the symmetric representation of sigma(m) has four parts.

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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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A365081 Numbers k with the property that the symmetric representation of sigma(k) has four parts and its second part is an octagon of width 1 and one of the vertices of the octagon is also the central vertex of the first valley of the largest Dyck path of the diagram.

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A264104 Numbers n with the property that the symmetric representation of sigma(n) has four parts, each of width one and two regions meet at the center of the Dyck path.

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