A346875 -id:A346875 - cp's OEIS frontend

A346864 Irregular triangle read by rows in which row n lists the row A014105(n) of A237591, n >= 1.

2, 1, 6, 2, 1, 1, 11, 4, 3, 1, 1, 1, 19, 6, 4, 2, 2, 1, 1, 1, 28, 10, 5, 3, 3, 2, 1, 1, 1, 1, 40, 13, 7, 5, 3, 2, 2, 2, 1, 1, 1, 1, 53, 18, 10, 5, 4, 3, 3, 2, 1, 2, 1, 1, 1, 1, 69, 23, 12, 7, 5, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 86, 29, 15, 9, 6, 5, 4, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Aug 17 2021

The characteristic shape of the symmetric representation of sigma(A014105(n)) consists in that in the main diagonal of the diagram the smallest Dyck path has a peak and the largest Dyck path has a valley.
So knowing this characteristic shape we can know if a number is a second hexagonal number (or not) just by looking at the diagram, even ignoring the concept of second hexagonal number.
Therefore we can see a geometric pattern of the distribution of the second hexagonal numbers in the stepped pyramid described in A245092.
T(n,k) is also the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A014105(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A014105(n).
T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th second hexagonal number into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th second hexagonal number into exactly k + 1 consecutive parts.
1 together with the first column gives A317186. - Michel Marcus, Jan 12 2025

			Triangle begins:
   2,  1;
   6,  2,  1, 1;
  11,  4,  3, 1, 1, 1;
  19,  6,  4, 2, 2, 1, 1, 1;
  28, 10,  5, 3, 3, 2, 1, 1, 1, 1;
  40, 13,  7, 5, 3, 2, 2, 2, 1, 1, 1, 1;
  53, 18, 10, 5, 4, 3, 3, 2, 1, 2, 1, 1, 1, 1;
  69, 23, 12, 7, 5, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1;
  86, 29, 15, 9, 6, 5, 4, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1;
...
Illustration of initial terms:
Column h gives the n-th second hexagonal number (A014105).
Column S gives the sum of the divisors of the second hexagonal numbers which equals the area (and the number of cells) of the associated diagram.
--------------------------------------------------------------------------------------
  n   h   S   Diagram
--------------------------------------------------------------------------------------
                  _             _                     _                             _
                 | |           | |                   | |                           | |
              _ _|_|           | |                   | |                           | |
  1   3   4  |_ _|1            | |                   | |                           | |
               2               | |                   | |                           | |
                            _ _| |                   | |                           | |
                           |  _ _|                   | |                           | |
                        _ _|_|                       | |                           | |
                       |  _|1                        | |                           | |
              _ _ _ _ _| | 1                         | |                           | |
  2  10  18  |_ _ _ _ _ _|2                          | |                           | |
                   6                          _ _ _ _|_|                           | |
                                             | |                                   | |
                                            _| |                                   | |
                                           |  _|                                   | |
                                        _ _|_|                                     | |
                                    _ _|  _|1                                      | |
                                   |_ _ _|1 1                                      | |
                                   |  3                               _ _ _ _ _ _ _| |
                                   |4                                |    _ _ _ _ _ _|
              _ _ _ _ _ _ _ _ _ _ _|                                 |   |
  3  21  32  |_ _ _ _ _ _ _ _ _ _ _|                              _ _|   |
                       11                                        |       |
                                                                _|    _ _|
                                                               |     |
                                                            _ _|    _|
                                                        _ _|      _|
                                                       |        _|1
                                                  _ _ _|    _ _|1 1
                                                 |         | 2
                                                 |  _ _ _ _|2
                                                 | |   4
                                                 | |
                                                 | |6
                                                 | |
              _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  4  36  91  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
                               19
.

Row sums give A014105, n >= 1.
Row lengths give A005843.
For the characteristic shape of sigma(A000040(n)) see A346871.
For the characteristic shape of sigma(A000079(n)) see A346872.
For the characteristic shape of sigma(A000217(n)) see A346873.
For the visualization of Mersenne numbers A000225 see A346874.
For the characteristic shape of sigma(A000384(n)) see A346875.
For the characteristic shape of sigma(A000396(n)) see A346876.
For the characteristic shape of sigma(A008588(n)) see A224613.
For the characteristic shape of sigma(A174973(n)) see A317305.
Cf. A000203, A237591, A237593, A245092, A249351, A262626, A317186.

row(n) = my(m=n*(2*n + 1)); vector((sqrtint(8*m+1)-1)\2, k, ceil((m+1)/k - (k+1)/2) - ceil((m+1)/(k+1) - (k+2)/2)); \\ Michel Marcus, Jan 12 2025

A350712 a(n) is the smallest hexagonal number for which the symmetric representation of sigma(n) has width 2*n, n >= 0, at the diagonal.

Original entry on oeis.org

0, 6, 120, 2016, 7140, 61776, 103740, 738720, 437580, 1185030, 4680270, 4426800, 2031120, 6193440, 4915680, 30728880, 2162160, 48565440, 134734320, 286071240, 163723560, 376902240, 536592420, 137373600, 76576500, 391986000, 214980480, 103672800, 1018606680, 5401294080

Offset: 0

Hartmut F. W. Hoft, Feb 02 2022

nonn

The width of the symmetric representation of sigma for hexagonal numbers at the diagonal is 1 only for number 1. For any hexagonal number h(n) = n*(2*n-1), n>1, the last leg of the Dyck path of h(n)-1 has length 2 and that of h(n) has length 1 (see formula in A237591) so that the width of the symmetric representation of sigma at the diagonal is at least 2 and contains a subpart of size 1 at the diagonal (see A280851).
The geometry of the Dyck paths ensures that a square bisected by the diagonal whose side length equals the width of the symmetric representation of sigma at the diagonal fits between the bounding pair of Dyck paths.
For hexagonal numbers up to h(100000) = 19999900000 only 1225, 1413721, and 1631432881 (the 25th, 841st, and 28561st hexagonal numbers) have width 3 at the diagonal, and none were found of odd width greater than 3.
The next [last] number in the sequence data smaller than h(55000) = 6049945000 is a(42) = 4874349480 [a(49) = 4819214400] with a(31..41) > h(55000).
The numbers [1, 1225, 1413721, 1631432881] mentioned above (in the first comment and in the third comment) are the first four square-hexagonal numbers (A046177). - Omar E. Pol, Feb 04 2022

			a(1) = 6, and a(2) = 120 since all hexagonal numbers k, 6 <= k < 120, have width 2 at the diagonal.

Cf. A000384, A046177, A235791, A237048, A237591, A237593, A249223, A280851, A346875.

(* for function a2[ ] see A237048 and A249223 *)
(* parameter bw is an upper bound estimate for how many values will be returned *)
a350712[n_, bw_] := Module[{widthL=Table[0, bw], wL, cL, i, w}, wL=Map[#(2#-1)&, Range[n]]; cL=Map[Last[a2[#]]&, wL]; For[i=1, i<=n, i++, w=cL[[i]]; If[EvenQ[w]&&widthL[[w/2]]==0, widthL[[w/2]]=wL[[i]]]]; Join[{0}, widthL]]
Take[a350712[55000, 50], 37]

A352015 Square array read by antidiagonals upwards: T(n,k) is the n-th number m such that the symmetric representation of sigma(m) has at least one subpart k, with n >= 1, k >= 1, m >= 1.

Original entry on oeis.org

1, 6, 3, 15, 18, 2, 28, 45, 5, 7, 45

Offset: 1

Omar E. Pol, Feb 28 2022

			The corner of the square array looks like this:
   1,  3,  2,  7, ...
   6, 18,  5, ...
  15, 45, ...
  28, ...
  ...
For n = 3 and k = 2 we have that 45 is the third positive integer m whose symmetric representation of sigma(m) has at least one subpart 2, so T(3,2) = 45.
For n = 5 and k = 1 we have that 45 is also the fifth positive integer m whose symmetric representation of sigma(m) has at least one subpart 1, so T(5,1) = 45.

Row 1 gives A351904.
Column 1 gives A000384.
Cf. A000203, A001227 (number of subparts), A196020, A235791, A236104, A237270, A237271, A237591, A237593, A279387 (definition of subparts), A280850, A280851 (subparts), A296508, A346875, A347529, A351819.

cp's OEIS Frontend

A346864 Irregular triangle read by rows in which row n lists the row A014105(n) of A237591, n >= 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A350712 a(n) is the smallest hexagonal number for which the symmetric representation of sigma(n) has width 2*n, n >= 0, at the diagonal.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A352015 Square array read by antidiagonals upwards: T(n,k) is the n-th number m such that the symmetric representation of sigma(m) has at least one subpart k, with n >= 1, k >= 1, m >= 1.

Views

Author

Keywords

Examples

Crossrefs