A280107 -id:A280107 - cp's OEIS frontend

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.

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

A264102 Numbers n with the property that the symmetric representation of sigma(n) has four parts, each of width one.

Original entry on oeis.org

21, 27, 33, 39, 51, 55, 57, 65, 69, 85, 87, 93, 95, 111, 115, 119, 123, 125, 129, 133, 141, 145, 155, 159, 161, 177, 183, 185, 201, 203, 205, 213, 215, 217, 219, 230, 235, 237, 249, 250, 253, 259, 265, 267, 287, 290, 291, 295, 301, 303, 305, 309, 310, 319, 321, 327, 329, 335

Offset: 1

Hartmut F. W. Hoft, Nov 03 2015

The areas of the first two regions are (2^(m+1) - 1) * (p * q + 1) / 2 and (2^(m+1) - 1) * (p + q) / 2, respectively. Twice their sum equals sigma(n) = (2^(m+1) - 1) * (p + 1) * (q + 1).

For a proof of the formula for this sequence see the link.

			65 = 5*13 is in the sequence since m = 0 and 2 < 5 < 10 < 13. The first two regions in the symmetric representation of sigma(65) = 84 start with legs 1 and 5 of the Dyck path and have areas 33 and 9, respectively.
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.
One case in the formula for the sequence is the 3-parameter expression n = 2^m * p * q with p and q distinct primes satisfying the stated conditions. That subsequence can be visualized as a skew tetrahedron since the start of each "line" on an irregular "triangular" side of the "tetrahedron" is determined by a different prime number and each layer is determined by a different power of two. Below are the first three layers with primes p designating columns and primes q rows.
m=0| 3    5    7    11   13
-----------------------------
7  | 21
11 | 33   55
13 | 39   65
17 | 51   85   119
19 | 57   95   133
23 | 69   115  161  253
29 | 87   145  203  319  377
31 | 93   155  217  341  403
37 | 111  185  259  407  481
41 | 123  205  287  451  533
...
89 | 267  445  623  979  1157
...
Column 1 is A001748 except for the first three terms and column 2 is A001750 except for the first four terms in the two resepctive sequences.
m=1| 3    5    7    11   13
-------------------------------
23 |     230
29 |     290  406
31 |     310  434
37 |     370  518
41 |     410  574
43 |     430  602
47 |     470  658  1034
53 |     530  742  1166  1378
...
89 |     890  1246 1958  2314
...
m=2| 3    5    7    11   13
-------------------------------
89 |               3916
97 |               4268
101|               4444
103|               4532
107|               4708  5564
109|               4796  5668
...
The fourth layer for m = 3 starts with number 37672 in column p = 17 and row q = 277.
The subsequence of the 2-parameter case n = 2^m * p^3 with 2^(m+1) < p gives rise to the following irregular triangle:
p\m| 0      1       2       3
----------------------------------
3  | 27
5  | 125    250
7  | 343    686
11 | 1331   2662    5324
13 | 2197   4394    8788
17 | 4913   9826    19652   39304
19 | 6859   13718   27436   54872
23 | 12167  24334   48668   97336
29 | 24389  48778   97556   195112
...
The first column in this triangle is A030078 except for the first term and the second column is A172190 except for the first two terms respectively in the two sequences.

Hartmut F. W. Hoft, Diagram of symmetric representations of sigma
Hartmut F. W. Hoft, Proof of formula for 4 regions of width 1

Cf. A001748, A001750, A030078, A172190.
For symmetric representation of sigma: A235791, A236104, A237270, A237271, A237591, A237593, A241008, A246955.
Subsequence of A280107.

mpStalk[m_, p_, bound_] := Module[{q=NextPrime[2^(m+1)*p], list={}}, While[2^m*p*q<=bound, AppendTo[list, 2^m*p*q]; q=NextPrime[q]]; If[2^m*p^3<=bound, AppendTo[list, 2^m*p^3]]; list]
mTriangle[m_, bound_] := Module[{p=NextPrime[2^(m+1)], list={}}, While[2^m*p*NextPrime[2^(m+1)*p]<=bound, list=Union[list, mpStalk[m, p, bound]]; p=NextPrime[p]]; list]
(* 2^(4m+3)<=bound is a simpler test, but computes some empty stalks *)
a264102[bound_] := Module[{m=0, list={}}, While[2^m*NextPrime[2^(m+1)]*NextPrime[2^(m+1)*NextPrime[2^(m+1)]]<=bound, list=Union[list, mTriangle[m, bound]]; m++]; list]
a264102[335] (* data *)

n = 2^m * p * q where m >= 0, p > 2 is prime, 2^(m+1) < p < 2^(m+1) * p < q, and either q is prime or q = p^2.

A320066 Numbers k with the property that the symmetric representation of sigma(k) has five parts.

Original entry on oeis.org

63, 81, 99, 117, 153, 165, 195, 231, 255, 273, 285, 325, 345, 375, 425, 435, 459, 475, 525, 561, 575, 625, 627, 665, 693, 725, 735, 775, 805, 819, 825, 875, 897, 925, 975, 1015, 1025, 1075, 1085, 1150, 1175, 1225, 1250, 1295, 1377, 1395, 1421, 1435, 1450, 1479, 1505, 1519, 1550, 1581, 1617, 1645, 1653, 1665

Offset: 1

Omar E. Pol, Oct 05 2018

nonn

Those numbers in this sequence with only parts of width 1 in their symmetric representation of sigma form column 5 in the table of A357581. - Hartmut F. W. Hoft, Oct 04 2022

			63 is in the sequence because the 63rd row of A237593 is [32, 11, 6, 4, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 4, 6, 11, 32], and the 62nd row of the same triangle is [32, 11, 5, 4, 3, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 3, 4, 5, 11, 32], therefore between both symmetric Dyck paths there are five parts: [32, 12, 16, 12, 32].
The sums of these parts is 32 + 12 + 16 + 12 + 32 = 104, equaling the sum of the divisors of 63: 1 + 3 + 7 + 9 + 21 + 63 = 104.
(The diagram of the symmetric representation of sigma(63) = 104 is too large to include.)

Column 5 of A240062.
Cf. A000203, A018267, A237270 (the parts), A237271 (number of parts), A174973 (one part), A239929 (two parts), A279102 (three parts), A280107 (four parts).
Cf. A236104, A237591, A237593, A239663, A239665, A245092, A262626.
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&]]
a320066[n_] := Select[Range[n], partsSRS[#]==5&]
a320066[1665] (* Hartmut F. W. Hoft, Oct 04 2022 *)

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 *)

A320511 Numbers k with the property that the symmetric representation of sigma(k) has six parts.

Original entry on oeis.org

147, 171, 189, 207, 243, 261, 275, 279, 297, 333, 351, 363, 369, 387, 423, 429, 465, 477, 507, 531, 549, 555, 595, 603, 605, 615, 639, 645, 657, 663, 705, 711, 715, 741, 747, 795, 801, 833, 845, 867, 873, 885, 909, 915, 927, 931, 935, 963, 969, 981, 1005, 1017, 1045, 1065, 1071, 1083, 1095, 1105, 1127

Offset: 1

Omar E. Pol, Oct 14 2018

nonn

Those numbers in this sequence with only parts of width 1 in their symmetric representation of sigma form column 6 in the table of A357581. - Hartmut F. W. Hoft, Oct 04 2022

			147 is in the sequence because the 147th row of A237593 is [74, 25, 13, 8, 5, 4, 4, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 4, 4, 5, 8, 13, 25, 74], and the 146th row of the same triangle is [74, 25, 12, 8, 6, 4, 3, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 4, 6, 8, 12, 25, 74], therefore between both symmetric Dyck paths there are six parts: [74, 26, 14, 14, 26, 74].
Note that the sum of these parts is 74 + 26 + 14 + 14 + 26 + 74 = 228, equaling the sum of the divisors of 147: 1 + 3 + 7 + 21 + 49 + 147 = 228.
(The diagram of the symmetric representation of sigma(147) = 228 is too large to include.)

Column 6 of A240062.
Cf. A237270 (the parts), A237271 (number of parts), A174973 (one part), A239929 (two parts), A279102 (three parts), A280107 (four parts), A320066 (five parts).
Cf. A000203, A018303, A196020, A235791, A236104, A237048, A237591, A237593, A239663, A239665, A245092, A262626, A296508.
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&]]
a320511[n_] := Select[Range[n], partsSRS[#]==6&]
a320511[1127] (* Hartmut F. W. Hoft, Oct 04 2022 *)

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

Original entry on oeis.org

10, 14, 22, 26, 34, 38, 44, 46, 52, 58, 62, 68, 74, 76, 78, 82, 86, 92, 94, 102, 106, 114, 116, 118, 122, 124, 134, 136, 138, 142, 146, 148, 152, 158, 164, 166, 172, 174, 178, 184, 186, 188, 194, 202, 206, 212, 214, 218, 222, 226, 232, 236, 244, 246, 248, 254, 258, 262, 268, 274, 278, 282, 284, 292, 296, 298, 302, 314, 316, 318, 326, 328, 332, 334, 344, 346, 348, 354, 356, 358

Offset: 1

Omar E. Pol, Jul 07 2014

nonn

Even numbers in A239929.

By definition the two parts of the symmetric representation of sigma(n) are sigma(n)/2 and sigma(n)/2.

			Illustration of the symmetric representation of sigma(n) in the second quadrant for the first four elements of this sequence: [10, 14, 22, 26].
.
.                             _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                            |  _ _ _ _ _ _ _ _ _ _ _ _ _|
.                            | |
.                            | |
.                            | |  _ _ _ _ _ _ _ _ _ _ _ _
.                      21 _ _| | |  _ _ _ _ _ _ _ _ _ _ _|
.                        |_ _ _| | |
.                     _ _|       | |
.                   _|     18 _ _| |
.                  |         |_ _ _|
.            21 _ _|        _|
.              | |        _|
.     _ _ _ _ _| | 18 _ _|                _ _ _ _ _ _ _ _
.    |  _ _ _ _ _|   | |                 |  _ _ _ _ _ _ _|
.    | |      _ _ _ _| |                 | |
.    | |     |  _ _ _ _|             12 _| |
.    | |     | |                       |_ _|  _ _ _ _ _ _
.    | |     | |                 12 _ _|     |  _ _ _ _ _|
.    | |     | |              _ _ _| |    9 _| |
.    | |     | |             |  _ _ _|  9 _|_ _|
.    | |     | |             | |      _ _| |
.    | |     | |             | |     |  _ _|
.    | |     | |             | |     | |
.    | |     | |             | |     | |
.    | |     | |             | |     | |
.    | |     | |             | |     | |
.    |_|     |_|             |_|     |_|
.
n:    26      22              14      10
.
Sigma(10) =  9 +  9 = 18.
Sigma(14) = 12 + 12 = 24.
Sigma(22) = 18 + 18 = 36.
Sigma(26) = 21 + 21 = 42.
.

R. J. Mathar, Table of n, a(n) for n = 1..999

Cf. A237271 (number of parts), A237270, A237593, A238443, A238524, A239660, A239929, A239932, A239934, A245092, A262626, A280107 (4 parts).

Extended by R. J. Mathar, Oct 04 2018

A298855 Squarefree semiprimes pq for which the symmetric representation of sigma(pq) has four parts, in increasing order.

Original entry on oeis.org

21, 33, 39, 51, 55, 57, 65, 69, 85, 87, 93, 95, 111, 115, 119, 123, 129, 133, 141, 145, 155, 159, 161, 177, 183, 185, 201, 203, 205, 213, 215, 217, 219, 235, 237, 249, 253, 259, 265, 267, 287, 291, 295, 301, 303, 305, 309, 319, 321, 327, 329, 335, 339, 341, 355, 365, 371, 377, 381, 393, 395

Offset: 1

Hartmut F. W. Hoft, Jan 27 2018

All numbers in this sequence are odd since the symmetric representation of 2*p, p prime > 3, has two parts each of size 3*(p+1)/2, and that for 6 has one part of size 12.
A number in this sequence has the form p*q, p and q prime, 3 <= p and 2*p < q, since in this case 2*p <= floor((sqrt(8*p*q + 1) - 1)/2) < q so that 1's in row p*q of A237048 occur only in positions 1, 2, p and 2*p.
This sequence is a subsequence of A046388, hence of A006881, as well as of A174905, A241008 and A280107.
The two central parts of the symmetric representation of sigma(p*q), each of size (p+q)/2, meet on the diagonal when q = 2*p + 1 since in this case 2*p = floor((sqrt(8*p*q + 1) - 1)/2). These triangular numbers p*(2p+1) form sequence A156592, except for its first element 10, and form a subsequence of the diagonal in the associated irregular triangle of this sequence given in the Example section. They also are a subsequence of A264104. A function to compute the coordinates on the diagonal where the two central parts meet is defined in sequence A240542.
Except for missing 10 the intersection of this sequence and A298856 equals A156592.

			21=3*7 is the smallest number in the sequence since 2*3<7.
1081=23*(2*23+1) is in the sequence; its central parts meet at 751 on the diagonal.
The semiprimes p*q can be arranged as an irregular triangle with rows and columns labeled by the respective odd primes:
  q\p|   3    5    7   11   13   17   19   23
  ---+---------------------------------------
   7 |  21
  11 |  33   55
  13 |  39   65
  17 |  51   85  119
  19 |  57   95  133
  23 |  69  115  161  253
  29 |  87  145  203  319  377
  31 |  93  155  217  341  403
  37 | 111  185  259  407  481  629
  41 | 123  205  287  451  533  697  779
  43 | 129  215  301  473  559  731  817
  47 | 141  235  329  517  611  799  893 1081

Cf. A001358, A005384, A005385, A006881, A046388, A068443, A156592, A174905, A237048, A237270, A237593, A240542, A241008, A264104, A280107, A298856.

(* Function a237270[] is defined in A237270 *)
a006881Q[n_] := Module[{f=FactorInteger[n]}, Length[f]==2 && AllTrue[Last[Transpose[f]], #==1&]]
a298855[m_, n_] := Select[Range[m, n], a006881Q[#] && Length[a237270[#]]==4 &]
a298855[1, 400] (* data *)
(* column for prime p through number n *)
stalk[n_, p_] := Select[a298855[1, n], First[First[FactorInteger[#]]]==p&]

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

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 14 2018

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

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

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

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

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

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.

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

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 04 2018

nonn

Also, even numbers of A239929 divided by two.

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

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

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

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

cp's OEIS Frontend

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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A264102 Numbers n with the property that the symmetric representation of sigma(n) has four parts, each of width one.

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A320066 Numbers k with the property that the symmetric representation of sigma(k) has five 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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A320511 Numbers k with the property that the symmetric representation of sigma(k) has six parts.

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

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A298855 Squarefree semiprimes p*q for which the symmetric representation of sigma(p*q) has four parts, in increasing order.

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

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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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A298855 Squarefree semiprimes pq for which the symmetric representation of sigma(pq) has four parts, in increasing order.