A250070 -id:A250070 - cp's OEIS frontend

A279391 Irregular triangle read by rows in which row n lists the subparts of the successive layers of the symmetric representation of sigma(n).

1, 3, 2, 2, 7, 3, 3, 11, 1, 4, 4, 15, 5, 3, 5, 9, 9, 6, 6, 23, 5, 7, 7, 12, 12, 8, 7, 8, 1, 31, 9, 9, 35, 2, 2, 10, 10, 39, 3, 11, 5, 5, 11, 18, 18, 12, 12, 47, 13, 13, 5, 13, 21, 21, 14, 6, 6, 14, 55, 1, 15, 15, 59, 3, 7, 3, 16, 16, 63, 17, 7, 7, 17, 27, 27, 18, 9, 18, 3, 71, 10, 10, 19, 19, 30, 30

Offset: 1

Omar E. Pol, Dec 12 2016

Note that the terms in the n-th row are the same as the terms in the n-th row of triangle A280851, but in some rows the terms appear in distinct order. First differs from A280851 at a(28) = T(15,3). - Omar E. Pol, Apr 24 2018

Row n in the triangle is a sequence of A250068(n) symmetric sections, each section consisting of the sizes of the subparts on that level in the symmetric representation of sigma of n - from the top down in the images below or left to right as drawn in A237593. - Hartmut F. W. Hoft, Sep 05 2021

			Triangle begins (first 15 rows):
   [1];
   [3];
   [2, 2];
   [7];
   [3, 3];
   [11], [1];
   [4, 4];
   [15];
   [5, 3, 5];
   [9, 9];
   [6, 6];
   [23], [5];
   [7, 7];
   [12, 12];
   [8, 7, 8], [1];
  ...
For n = 12 we have that 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"
.   the 12th row of the                that contain 23 and 5 cells
.   triangle A237270 is [28].          respectively, so the 12th row of
.                                      this triangle is [23], [5].
.
For n = 15 we have that 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 the 15th row of        The first layer has three subparts:
.   triangle A237270 is [8, 8, 8].     8, 7, 8. The second layer has
.                                      only one subpart of size 1, so
.                                      the 15th row of this triangle is
.                                      [8, 7, 8], [1].
.
The smallest even number with 3 levels is 60; its row of subparts is: [119], [37], [6, 6]. The smallest odd number with 3 levels is 315; its row of subparts is:  [158, 207, 158], [11, 26, 5, 9, 5, 26, 11], [4, 4]. - _Hartmut F. W. Hoft_, Sep 05 2021

Index entries for sequences related to sigma(n)

The length of row n equals A001227(n).
If n is odd the length of row n equals A000005(n).
Row sums give A000203.
For the definition of "subparts" see A279387.
For the triangle of sums of subparts see A279388.
Cf. A001227, A005279, A196020, A235791, A236104, A237048, A237270, A237591, A237593, A239657, A244050, A245092, A250068, A250070, A261699, A262626, A280851, A296508.

(* support functions are defined in aA237593 and A262045 *)
subP[level_] := Module[{s=Map[Apply[Plus, #]&, Select[level, First[#]!=0&]]}, If[OddQ[Length[s]], s[[(Length[s]+1)/2]]-=1]; s]
a279391[n_] := Module[{widL=a262045[n], lenL=a237593[n], srs, subs}, srs=Transpose[Map[PadRight[If[widL[[#]]>0, Table[1, widL[[#]]], {0}], Max[widL]]&, Range[Length[lenL]]]]; subs=Map[SplitBy[lenL srs[[#]], #!=0&]&, Range[Max[widL]]]; Flatten[Map[subP, subs]]]
Flatten[Map[a279391, Range[38]]] (* Hartmut F. W. Hoft, Sep 05 2021 *)

A250071 Smallest number k such that the symmetric representation of sigma(k) has maximum width n for those k whose representation has nondecreasing width up to the diagonal.

Original entry on oeis.org

1, 6, 72, 120, 5184, 1440, 373248, 6720, 28800, 103680, 1934917632, 80640, 278628139008, 7464960, 2073600, 483840, 1444408272617472, 1612800, 103997395628457984, 5806080, 298598400, 77396705280, 539122498937926189056, 7096320, 1658880000, 5572562780160, 90316800, 418037760, 402452788967166148425547776, 116121600

Offset: 1

Hartmut F. W. Hoft, Nov 11 2014

nonn

The symmetric representation of sigma(k) has nondecreasing width to the diagonal precisely when all odd divisors counted in the k-th row of A237048 occur at odd indices. If we write k = 2^m * q with m >= 0 and q odd, this property is equivalent to q < 2^(m+1).
The values for a(11), a(13), a(17) and a(19) were computed directly using the formula k = 2^m * 3^(p-1) where p is one of the four primes and m the smallest exponent so that 3^(p-1) < 2^(m+1). Each of these numbers has a symmetric representation of nondecreasing width ending in a prime number width, and they are the first such numbers since the number of divisors of an odd number is a prime precisely when the number is a power of a prime.
The other numbers listed whose symmetric representations of sigma(k) have nondecreasing width are smaller than 7500000. The only additional numbers k <= 100000000 are a(24) = 7096320, a(27) = 90316800 and a(32) = 85155840.
See A340506 for another way to look at this data. - N. J. A. Sloane, Jan 23 2021

			a(6) = 1440 = 2^5 * 3^2 * 5 has 6 odd divisors. It is the smallest number of the form 2^m * q with m > 0, q odd and such that q < 2^(m+1).

Cf. A000203, A237048, A237270, A237271, A237593, A241008, A241010, A246955, A247687, A249223, A250068, A250070.
Cf. also A340506.
Cf. A162247.

(* function a2[ ] is defined in A249223 *)
smallQ[n_] := Module[{x=2^IntegerExponent[n,2]}, n/x<2x]
ndWidth[{m_,n_}] := Select[Range[m, n], smallQ]
a250071[x_List] := Module[{i, max, acc={{1, 1}}}, For[i=1, i<=Length[x], i++, max={Max[a2[x[[i]]]], x[[i]]}; If[!MemberQ[Transpose[acc][[1]], max[[1]]], AppendTo[acc, max]]]; acc]
(* returns (argument,result) data pairs since sequence is non-monotonic *)
Sort[a250071[ndWidth[{1,100000000}]]] (* computed in steps *)
(* alternate implementation using function f[ ] by T. D. Noe in A162247 *)
sF[n_] := Min[Map[Apply[Times, Prime[Range[2, Length[#]+1]]^#]&, Map[Reverse[#-1]&, f[n]]]]
f1U[n_] := Module[{s=sF[n], k}, k=Floor[Log[2, s]]; 2^k s]
a250071[n_] := Map[f1U, Range[n]]
a250071[30] (* Hartmut F. W. Hoft, Nov 27 2024 *)

a(n) = min(2^m * q, m >= 0 & q odd & sigma_0(q) = n & q < 2^(m+1)) where sigma_0 is the number of divisors.

a(p) = 2^ceiling((p-1)*(log_2(3)) - 1) * 3^(p-1) for primes p.

a(21)-a(30) from Hartmut F. W. Hoft, Nov 27 2024

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

Original entry on oeis.org

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

Offset: 1

Hartmut F. W. Hoft, Aug 07 2014

nonn

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

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

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

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

Cf. A318843

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

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

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

A279388 Irregular triangle read by rows: T(n,k) is the sum of the subparts in the k-th layer of the symmetric representation of sigma(n), if such a layer exists.

Original entry on oeis.org

1, 3, 4, 7, 6, 11, 1, 8, 15, 13, 18, 12, 23, 5, 14, 24, 23, 1, 31, 18, 35, 4, 20, 39, 3, 32, 36, 24, 47, 13, 31, 42, 40, 55, 1, 30, 59, 13, 32, 63, 48, 54, 45, 3, 71, 20, 38, 60, 56, 79, 11, 42, 83, 13, 44, 84, 73, 5, 72, 48, 95, 29, 57, 93, 72, 98, 54, 107, 13, 72, 111, 9, 80, 90, 60, 119, 37, 12

Offset: 1

Omar E. Pol, Dec 12 2016

			Triangle begins (first 15 rows):
  1;
  3;
  4;
  7;
  6;
  11, 1;
  8;
  15;
  13;
  18;
  12;
  23, 5;
  14;
  24;
  23, 1;
  ...
For n = 12 we have that 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  _|    _ _|                       23  _|  _ _ _|
.                _|     |                             _|  _| |
.               |      _|                            |  _|  _|
.               |  _ _|                              | |_ _|
.    _ _ _ _ _ _| |                       _ _ _ _ _ _| |      5
.   |_ _ _ _ _ _ _|                      |_ _ _ _ _ _ _|
.
.   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"
.   A000203(12) = 28.                  that contain 23 and 5 cells
.                                      respectively, so the 12th row of
.                                      this triangle is [23, 5].
.
For n = 15 we have that 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) is constructed as shown below in Figure 3:
.                                _                                  _
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                           8   | |                            8   | |
.                               | |                                | |
.                               | |                                | |
.                          _ _ _|_|                           _ _ _|_|
.                   8  _ _| |                          7  _ _| |
.                     |    _|                            |  _ _|
.                    _|  _|                             _| |_|
.                   |_ _|                              |_ _|  1
.           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
.   whose sum is 8 + 8 + 8 = 24,       width 1 we can see four "subparts".
.   so A000203(15) = 24.               The first layer has three subparts
.                                      whose sum is 8 + 7 + 8 = 23. The
.                                      second layer has only one subpart
.                                      of size 1, so the 15th row of this
.                                      triangle is [23, 1].
.

Index entries for sequences related to sigma(n)

For the definition of "subparts" see A279387.
For the triangle of subparts see A279391.
Row sums give A000203.
Row n has length A250068(n).
Cf. A001227, A005279, A196020, A235791, A236104, A237048, A237270, A237591, A237593, A239657, A244050, A245092, A250070, A261699, A262626.

A262611 Triangle read by rows in which row n lists the widths of the symmetric representation of A024916(n): the sum of all divisors of all positive integers <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 3, 3, 3, 2, 1, 1, 2, 3, 3, 3, 3, 3, 2, 1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1, 2, 3, 4, 4, 4, 5, 4, 4, 4, 3, 2, 1, 1, 2, 3, 4, 5, 5, 5, 6, 5, 5, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 5, 5, 6, 7, 6, 5, 5, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 6, 6, 7, 7, 7, 6, 6, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 6, 6, 6, 7, 7, 7, 6, 6, 6, 6, 5, 4, 3, 2, 1

Offset: 1

Omar E. Pol, Sep 26 2015

Here T(n,k) is defined to be the "k-th width" of the symmetric representation of A024916(n), with n>=1 and 1<=k<=2n-1.
If both A249351 and this sequence are written as isosceles triangles then the partial sums of the columns of A249351 give the columns of this isosceles triangle (see the second triangle in Example section).
For the definition of the k-th width of the symmetric representation of sigma(n) see A249351.
Note that for the geometric representation of the n-th row of the triangle we need the x-axis, the y-axis, and only a Dyck path which is given by the elements of the n-th row of the triangle A237593.
Row n has length 2*n-1.
Row sums give A024916.
The middle diagonal is A240542.

			Triangle begins:
1;
1,2,1;
1,2,2,2,1;
1,2,3,3,3,2,1;
1,2,3,3,3,3,3,2,1;
1,2,3,4,4,5,4,4,3,2,1;
1,2,3,4,4,4,5,4,4,4,3,2,1;
1,2,3,4,5,5,5,6,5,5,5,4,3,2,1;
1,2,3,4,5,5,5,6,7,6,5,5,5,4,3,2,1;
1,2,3,4,5,6,6,6,7,7,7,6,6,6,5,4,3,2,1;
1,2,3,4,5,6,6,6,6,7,7,7,6,6,6,6,5,4,3,2,1;
1,2,3,4,5,6,7,7,7,8,9,9,9,8,7,7,7,6,5,4,3,2,1;
...
--------------------------------------------------------------------------
.        Written as an isosceles triangle
.              the sequence begins:               Diagram for n = 1..12
--------------------------------------------------------------------------
.                                                _ _ _ _ _ _ _ _ _ _ _ _
.                      1;                       |_| | | | | | | | | | | |
.                    1,2,1;                     |_ _|_| | | | | | | | | |
.                  1,2,2,2,1;                   |_ _|  _|_| | | | | | | |
.                1,2,3,3,3,2,1;                 |_ _ _|    _|_| | | | | |
.              1,2,3,3,3,3,3,2,1;               |_ _ _|  _|  _ _|_| | | |
.            1,2,3,4,4,5,4,4,3,2,1;             |_ _ _ _|  _| |  _ _|_| |
.          1,2,3,4,4,4,5,4,4,4,3,2,1;           |_ _ _ _| |_ _|_|    _ _|
.        1,2,3,4,5,5,5,6,5,5,5,4,3,2,1;         |_ _ _ _ _|  _|     |
.      1,2,3,4,5,5,5,6,7,6,5,5,5,4,3,2,1;       |_ _ _ _ _| |      _|
.    1,2,3,4,5,6,6,6,7,7,7,6,6,6,5,4,3,2,1;     |_ _ _ _ _ _|  _ _|
.  1,2,3,4,5,6,6,6,6,7,7,7,6,6,6,6,5,4,3,2,1;   |_ _ _ _ _ _| |
.1,2,3,4,5,6,7,7,7,8,9,9,9,8,7,7,7,6,5,4,3,2,1; |_ _ _ _ _ _ _|
...
For n = 3 the symmetric representation of A024916(3) = 8 in the 4th quadrant looks like this:
.
.    Polygon         Cells
.     _ _ _          _ _ _
.    |     |        |_|_|_|
.    |    _|        |_|_|_|
.    |_ _|          |_|_|
.
There are eight cells. The representation of the widths looks like this:
.
.     \ \ \
.     \ \ \
.     \ \    1
.          2 2
.        1 2
.
So the third row of the triangle is [1, 2, 2, 2, 1].

Cf. A024916, A236104, A237048, A237593, A240542, A241008, A241010, A244250, A245685, A246955, A247687, A249351, A250068, A250070, A250071, A253258, A262626.

A253258 Square array read by antidiagonals, j>=1, k>=1: T(j,k) is the j-th number n such that the symmetric representation of sigma(n) has at least a part with maximum width k.

Original entry on oeis.org

1, 2, 6, 3, 12, 60, 4, 15, 72, 120, 5, 18, 84, 180, 360, 7, 20, 90, 240, 420, 840, 8, 24, 126, 252, 720, 1080, 3360, 9, 28, 140, 336, 1008, 1260, 3600, 2520, 10, 30, 144, 378, 1200, 1440, 3780, 5544, 5040, 11, 35, 168, 432, 1320, 1680, 3960, 6300, 7560, 10080, 13, 36, 198, 480, 1512, 1800, 4200, 6720, 9240, 12600, 15120

Offset: 1

Omar E. Pol, Jul 08 2015

This is a permutation of the natural numbers.
Row 1 gives A250070.
For more information about the widths of the symmetric representation of sigma see A249351 and A250068.
The next term: 120 < T(2,4) < 360.
From Hartmut F. W. Hoft, Sep 20 2024: (Start)
Column T(j,1), j>=1, forms A174905 and is a permutation of A357581. Numbers T(j,k), j>=1 and k>1, form A005279. Conjecture: Every column of the square array contains odd numbers.
The sequence of smallest odd numbers in each column forms A347980. E.g., in column 12 the smallest odd number is T(466, 12) = 765765 = A347980(12) which is equivalent to A250068(765765) = 12. (End)

			The corner of the square array T(j,k) begins:
  1,  6, 60, 120, 360, ...
  2, 12, 72, ...
  3, 15, 84, ...
  4, 18, ...
  5, 20, ...
  7, ...
  ...
For j = 1 and k = 2; T(1,2) is the first number n such that the symmetric representation of sigma(n) has a part with maximum width 2 as shown below:
.
      Dyck paths            Cells              Widths
      _ _ _ _             _ _ _ _
      _ _ _  |_          |_|_|_|_|_          / / / /
           |   |_              |_|_|_              / /
           |_ _  |             |_|_|_|             / / /
               | |                 |_|                 /
               | |                 |_|                 /
               | |                 |_|                 /
.
The widths of the symmetric representation of sigma(6) = 12 are [1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1], also the 6th row of triangle A249351.
From _Hartmut F. W. Hoft_, Sep 20 2024: (Start)
Extending the terms T(j,k) to a 12x12 square array:
j\k 1  2  3   4   5    6    7    8     9     10    11    12
--------------------------------------------------------------
1 | 1  6  60  120 360  840  3360 2520  5040  10080 15120 32760
2 | 2  12 72  180 420  1080 3600 5544  7560  12600 20160 36960
3 | 3  15 84  240 720  1260 3780 6300  9240  13860 25200 39600
4 | 4  18 90  252 1008 1440 3960 6720  10920 15840 35280 41580
5 | 5  20 126 336 1200 1680 4200 6930  11880 16380 40320 43680
6 | 7  24 140 378 1320 1800 4320 7140  14040 16800 42840 45360
7 | 8  28 144 432 1512 1980 4620 7920  16632 18480 46800 46200
8 | 9  30 168 480 1560 2016 4680 8190  17160 18900 47880 47520
9 | 10 35 198 504 1848 2100 5280 8400  17640 21420 56160 49140
10| 11 36 210 540 1890 2160 5400 9360  18720 21840 56700 51480
11| 13 40 216 594 2184 2340 5460 10296 19800 22680 57120 52920
12| 14 42 264 600 2310 2640 5940 10800 20790 23760 57960 54600
...
(End)

Hartmut F. W. Hoft, Table of n, a(n) for n = 1..820
Index entries for sequences that are permutations of the natural numbers

Cf. A000203, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A249351, A250068, A250070, A250071.

Cf. A005279, A174905, A347980, A357581.

(* Computing table T(j,k) of size mxn with bound b *)
eP[n_] := If[EvenQ[n], FactorInteger[n][[1, 2]], 0]+1
sDiv[n_] := Module[{d=Select[Divisors[n], OddQ]}, Select[Union[d, d*2^eP[n]], #<=row[n]&]]
mWidth[n_] :=Max[FoldList[#1+If[OddQ[#2], 1, -1]&, sDiv[n]]]
t253258[{m_, n_}, b_] := Module[{s=Table[0, {i, m+1}, {j, n}], k=1, w, f}, While[k<=b, w=mWidth[k]; If[w<=n, f=s[[m+1, w]]; If[fHartmut F. W. Hoft, Sep 20 2024 *)

More terms from Charlie Neder, Jan 11 2019

A347980 a(n) is the smallest odd number k whose symmetric representation of sigma(k) has maximum width n.

Original entry on oeis.org

1, 15, 315, 2145, 3465, 17325, 45045, 51975, 225225, 405405, 315315, 765765, 1576575, 2297295

Offset: 1

Hartmut F. W. Hoft, Sep 22 2021

The sequence is not increasing with the maximum width of the symmetric representation just like A347979.

Observation: a(2)..a(14) ending in 5. - Omar E. Pol, Sep 23 2021

			The pattern of maximum widths of the parts in the symmetric representation of sigma for the first four terms in the sequence is:
   a(n) parts  successive widths
     1:   1          1
    15:   3        1 2 1
   315:   3        1 3 1
  2145:   7    1 2 3 4 3 2 1

Cf. A174973, A237048, A237270, A237271, A237591, A237593, A238443, A249351 (widths), A250070, A262045, A341969, A341970, A341971, A347979.

a262045[n_] := Module[{a=Accumulate[Map[If[Mod[n - # (#+1)/2, #]==0, (-1)^(#+1), 0] &, Range[Floor[(Sqrt[8n+1]-1)/2]]]]}, Join[a, Reverse[a]]]
a347980[n_, mw_] := Module[{list=Table[0, mw], i, v}, For[i=1, i<=n, i+=2, v=Max[a262045[i]]; If [list[[v]]==0, list[[v]]=i]]; list]
a347980[2500000,14] (* long evaluation time *)

A340506 For those rows n of A249223 which are weakly increasing, let w(n) denote the maximal entry in the row: sequence gives values of n for which w(n) sets a new record.

Original entry on oeis.org

1, 6, 72, 120, 1440, 6720, 28800, 80640, 483840, 1612800, 5806080, 7096320, 85155840, 283852800, 510935040, 1476034560, 7947878400, 17712414720, 29520691200, 106274488320, 354248294400, 1653158707200, 2125489766400, 4817776803840, 8029628006400, 28906660823040

Offset: 1

N. J. A. Sloane, Jan 23 2021

This is a companion to A250071 (and is derived from the data for that sequence), which lists the first time k appears as a width.
The record values are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, but more data is needed to identify this sequence.
The odd part of a(n) is A053624(n), n>=1. The record values 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, ... are the beginning of A053640. - Hartmut F. W. Hoft, Mar 29 2022

			a(4) = 120 = 2^3 * A053624(4) = 2^3 * 15 and a(7) = 28800 = 2^7 * A053624(7) = 2^7 * 225. - _Hartmut F. W. Hoft_, Mar 29 2022

Cf. A237270, A237593, A249223, A250068, A250070, A250071.

Cf. A053624, A053640.

prevPower2[k_] := If[k==1, 1, 2^(Ceiling[Log[2, k]]-1)]
a340506[n_] := Module[{recL={{1, 1}}, q, d, pp}, For[q=1, q<=n, q+=2, d=DivisorSigma[0, q]; pp=prevPower2[q] q; If[First[Last[recL]]Hartmut F. W. Hoft, Mar 29 2022 *)

a(n) = 2^t(n) * A053624(n), n > 1, where t(n) is the largest exponent satisfying 2^t(n) < A053624(n) and A053624(n) is the odd part of a(n) - see the comment in A250071. - Hartmut F. W. Hoft, Mar 29 2022

a(12)-a(26) from Hartmut F. W. Hoft, Mar 29 2022

A347979 a(n) is the smallest even number k whose symmetric representation of sigma(k) has maximum width n.

Original entry on oeis.org

2, 6, 60, 120, 360, 840, 3360, 2520, 5040, 10080, 15120, 32760, 27720, 50400, 98280, 83160, 110880, 138600, 221760, 277200, 332640, 360360, 554400, 960960, 831600, 942480, 720720, 2217600, 1965600, 1441440

Offset: 1

Hartmut F. W. Hoft, Sep 22 2021

For the 30 known terms the symmetric representation of sigma consists of a single part, i.e., this is a subsequence of A174973 = A238443.
The sequence is not increasing with the maximum width of the symmetric representation of sigma.
Also a(33) = 2162160 is the only further number in the sequence less than 2500000.

			The pattern of maximum widths within the single part of the symmetric representation of sigma for the first four numbers in the sequence is:
  a(n) parts successive widths
    2:   1           1
    6:   1         1 2 1
   60:   1     1 2 3 2 3 2 1
  120:   1     1 2 3 4 3 2 1

Cf. A174973, A237048, A237270, A237271, A237591, A237593, A238443, A249351 (widths), A250070, A262045, A341969, A341970, A341971, A347980.

a262045[n_] := Module[{a=Accumulate[Map[If[Mod[n - # (#+1)/2, #]==0, (-1)^(#+1), 0] &, Range[Floor[(Sqrt[8n+1]-1)/2]]]]}, Join[a, Reverse[a]]]
a347979[n_, mw_] := Module[{list=Table[0, mw], i, v}, For[i=2, i<=n, i+=2, v=Max[a262045[i]]; If [list[[v]]==0, list[[v]]=i]]; list]
a347979[2500000, 33] (* computes a(1..30), a(33); a(31..32) > 2500000 *)

It appears that a(n) = A250070(n) if n >= 2.

A348171 Square array read by upward antidiagonals in which T(w,p) is the smallest number k whose symmetric representation of sigma(k) consists of p parts with maximum width w occurring in at least one of its p parts.

Original entry on oeis.org

1, 6, 3, 60, 78, 9, 120, 7620, 15, 21, 360, 28920, 315, 75, 81, 840, 261720, 1326, 495, 63, 147, 3360, 1422120, 3465, 22542, 525, 189, 729, 2520, 22622880, 17325, 44574, 5005, 1275, 357, 903, 5040, 12728520, 45045, 199578, 6435, 16575, 1287, 1197, 3025, 10080, 50858640, 51975, 7734558, 34034, 131835, 2145, 3861, 2499, 6875

Offset: 1

Hartmut F. W. Hoft, Oct 04 2021

The first row of the table below is A318843 and the first column is A250070.
T(1,k+1) <= 3^k, for all k>=0, since for k=2j the (j+1)-st part in the symmetric representation of sigma(3^k) extends across the diagonal, and for k=2j+1 the (j+1)-st part is completed before the diagonal.
The data computed so far for a partially filled table of 15 rows and 15 columns, show that all rows, all columns (except column 4 for n <= 6 *10^7), and the diagonal are nonmonotonic.

			The 10x10 section of the table with dashes indicating values greater than 6*10^7; rows w denote the maximum width and columns p the number of parts in the symmetric representation of sigma(T(w,p)).
w\p | 1     2        3      4       5       6       7       8        9   ...
----------------------------------------------------------------------------
  1 | 1     3        9      21      81      147     729     903      3025
  2 | 6     78       15     75      63      189     357     1197     2499
  3 | 60    7620     315    495     525     1275    1287    3861     3591
  4 | 120   28920    1326   22542   5005    16575   2145    29325    11583
  5 | 360   261720   3465   44574   6435    131835  76125   24225    82593
  6 | 840   1422120  17325  199578  34034   83655   196707  468027   62985
  7 | 3360  22622880 45045  7734558 153153  442442  314925  1108965  471975
  8 | 2520  12728520 51975     -    205275  2067065 1429275 2359875  557175
  9 | 5040  50858640 225225    -    646646  2863718 2395197 5353725  2785875
  10| 10080    -     405405    -    1990989 2124694 6500375 36535499 7753875
   ...
The symmetric representation of sigma for T(2,3) = 15 consists of the three parts (8, 8, 8) of maximum widths (1, 2, 1), and that of T(3,3) = 315 consists of the three parts (158, 308, 158) of maximum widths (1, 3, 1).

Cf. A237048, A237270, A237271, A237591, A237593, A238443, A239663, A249223, A250070, A262045, A318843, A341969, A341970, A341971, A347979, A347980, A348142.

(* function a341969 is defined in A341969 *)
a348171[n_,  {w_, p_}] := Module[{list=Table[0, {i, w}, {j, p}], k, s, c, u}, For[k=1, k<=n, k++, s=Map[Max, Select[SplitBy[a341969[k], # != 0 &], #[[1]] != 0 &]]; c = Length[s]; u = Max[s]; If[u<=w && c<=p, If[list[[u, c]] == 0, list[[u, c]] = k ]]]; list]
table=a348171[60000000, {15, 15}] (* 15x15 table; very long computation time *)
p[n_] := n-row[n-1](row[n-1]+1)/2
w[n_] := row[n-1]-p[n]+2
Map[table[[w[#], p[#]]]&, Range[55]] (* sequence data *)

a((w+p-2)(w+p-1)/2 + p) = T(w,p), for all w, p >= 1.

T(w(n), p(n)) = a(n), for all n >= 1, where p(n) = n - r(n-1) * (r(n-1) + 1)/2, w(n) = r(n-1) - p(n) + 2, and r(n) = floor((sqrt(8*n+1) - 1)/2).

cp's OEIS Frontend

A279391 Irregular triangle read by rows in which row n lists the subparts of the successive layers of the symmetric representation of sigma(n).

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A250071 Smallest number k such that the symmetric representation of sigma(k) has maximum width n for those k whose representation has nondecreasing width up to the diagonal.

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

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A279388 Irregular triangle read by rows: T(n,k) is the sum of the subparts in the k-th layer of the symmetric representation of sigma(n), if such a layer exists.

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A262611 Triangle read by rows in which row n lists the widths of the symmetric representation of A024916(n): the sum of all divisors of all positive integers <= n.

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A253258 Square array read by antidiagonals, j>=1, k>=1: T(j,k) is the j-th number n such that the symmetric representation of sigma(n) has at least a part with maximum width k.

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A347980 a(n) is the smallest odd number k whose symmetric representation of sigma(k) has maximum width n.

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A340506 For those rows n of A249223 which are weakly increasing, let w(n) denote the maximal entry in the row: sequence gives values of n for which w(n) sets a new record.

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A347979 a(n) is the smallest even number k whose symmetric representation of sigma(k) has maximum width n.