A224613 -id:A224613 - 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

A346865 Sum of divisors of the n-th hexagonal number.

Original entry on oeis.org

1, 12, 24, 56, 78, 144, 112, 360, 234, 360, 384, 672, 434, 960, 720, 992, 864, 1872, 760, 2352, 1344, 1584, 1872, 2880, 1767, 3024, 2160, 4032, 2400, 4320, 1984, 6552, 4032, 3672, 4608, 6552, 2812, 7440, 5376, 7200, 5082, 8064, 4752, 10080, 7020, 8064, 6144

Offset: 1

Omar E. Pol, Aug 17 2021

nonn

The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram the smallest Dyck path has a valley and the largest Dyck path has a peak.
So knowing this characteristic shape we can know if a number is an hexagonal number (or not) just by looking at the diagram, even ignoring the concept of hexagonal number.
Therefore we can see a geometric pattern of the distribution of the hexagonal numbers in the stepped pyramid described in A245092.

			a(3) = 24 because the sum of divisors of the third hexagonal number (i.e., 15) is 1 + 3 + 5 + 15 = 24.
On the other hand we can see that in the main diagonal of every diagram the smallest Dyck path has a valley and the largest Dyck path has a peak as shown below.
Illustration of initial terms:
-------------------------------------------------------------------------
  n  H(n)  a(n)  Diagram
-------------------------------------------------------------------------
                 _         _                 _                         _
  1    1    1   |_|       | |               | |                       | |
                          | |               | |                       | |
                       _ _| |               | |                       | |
                      |    _|               | |                       | |
                 _ _ _|  _|                 | |                       | |
  2    6   12   |_ _ _ _|                   | |                       | |
                                            | |                       | |
                                       _ _ _|_|                       | |
                                   _ _| |                             | |
                                  |    _|                             | |
                                 _|  _|                               | |
                                |_ _|                                 | |
                                |                                     | |
                 _ _ _ _ _ _ _ _|                            _ _ _ _ _| |
  3   15   24   |_ _ _ _ _ _ _ _|                           |  _ _ _ _ _|
                                                            | |
                                                         _ _| |
                                                     _ _|  _ _|
                                                    |    _|
                                                   _|  _|
                                                  |  _|
                                             _ _ _| |
                                            |  _ _ _|
                                            | |
                                            | |
                                            | |
                 _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  4   28   56   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Column H gives the nonzero hexagonal numbers (A000384).
a(n) is also the area (and the number of cells) of the n-th diagram.
For n = 3 the sum of the regions (or parts) of the third diagram is 8 + 8 + 8 = 24, so a(3) = 24.
For more information see A237593.

Bisection of A074285.
Cf. A000203, A000384, A237591, A237593, A245092, A262626, A346864.
Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346866 (of second hexagonal numbers), A346867 (of numbers with middle divisors), A346868 (of numbers with no middle divisors).

a[n_] := DivisorSigma[1, n*(2*n - 1)]; Array[a, 50] (* Amiram Eldar, Aug 18 2021 *)

a(n) = sigma(n*(2*n-1)); \\ Michel Marcus, Aug 18 2021

from sympy import divisors
def a(n): return sum(divisors(n*(2*n - 1)))
print([a(n) for n in range(1, 48)]) # Michael S. Branicky, Aug 20 2021

a(n) = A000203(A000384(n)).

Sum_{k=1..n} a(k) ~ 4*n^3/3. - Vaclav Kotesovec, Aug 18 2021

A346866 Sum of divisors of the n-th second hexagonal number.

Original entry on oeis.org

4, 18, 32, 91, 72, 168, 192, 270, 260, 576, 288, 868, 560, 720, 768, 1488, 864, 1482, 1120, 1764, 1408, 2808, 1152, 3420, 2232, 2268, 2880, 4480, 1800, 4464, 3328, 5292, 3264, 5184, 3456, 6734, 4712, 5760, 4480, 10890, 3528, 10368, 5280, 7560, 8736, 9216, 5760, 12152

Offset: 1

Omar E. Pol, Aug 17 2021

The characteristic shape of the symmetric representation of a(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.

			a(3) = 32 because the sum of divisors of the third second hexagonal number (i.e., 21) is 1 + 3 + 7 + 21 = 32.
On the other hand we can see that in the main diagonal of every diagram the smallest Dyck path has a peak and the largest Dyck path has a valley as shown below.
Illustration of initial terms:
---------------------------------------------------------------------------------------
  n  h(n)  a(n)  Diagram
---------------------------------------------------------------------------------------
                    _             _                     _                            _
                   | |           | |                   | |                          | |
                _ _|_|           | |                   | |                          | |
  1    3    4  |_ _|             | |                   | |                          | |
                                 | |                   | |                          | |
                              _ _| |                   | |                          | |
                             |  _ _|                   | |                          | |
                          _ _|_|                       | |                          | |
                         |  _|                         | |                          | |
                _ _ _ _ _| |                           | |                          | |
  2   10   18  |_ _ _ _ _ _|                           | |                          | |
                                                _ _ _ _|_|                          | |
                                               | |                                  | |
                                              _| |                                  | |
                                             |  _|                                  | |
                                          _ _|_|                                    | |
                                      _ _|  _|                                      | |
                                     |_ _ _|                                        | |
                                     |                                 _ _ _ _ _ _ _| |
                                     |                                |    _ _ _ _ _ _|
                _ _ _ _ _ _ _ _ _ _ _|                                |   |
  3   21   32  |_ _ _ _ _ _ _ _ _ _ _|                             _ _|   |
                                                                  |       |
                                                                 _|    _ _|
                                                                |     |
                                                             _ _|    _|
                                                         _ _|      _|
                                                        |        _|
                                                   _ _ _|    _ _|
                                                  |         |
                                                  |  _ _ _ _|
                                                  | |
                                                  | |
                                                  | |
                                                  | |
               _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  4   36   91 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Column h gives the n-th second hexagonal number (A014105).
The widths of the main diagonal of the diagrams are [0, 0, 0, 1] respectively.
a(n) is the area (and the number of cells) of the n-th diagram.
For n = 3 the sum of the regions (or parts) of the third diagram is 11 + 5 + 5 + 11 = 32, so a(3) = 32.
For n = 4 there is only one region (or part) of size 91 in the fourth diagram so a(4) = 91.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Bisection of A074285.
Cf. A000203, A000384, A014105, A237591, A237593, A245092, A262626, A346864.
Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346865 (of hexagonal numbers), A346867 (of numbers with middle divisors), A346868 (of numbers with no middle divisors), A347155 (of nontriangular numbers).

a[n_] := DivisorSigma[1, n*(2*n + 1)]; Array[a, 50] (* Amiram Eldar, Aug 18 2021 *)

a(n) = sigma(n*(2*n + 1)); \\ Michel Marcus, Aug 18 2021

a(n) = A000203(A014105(n)).

Sum_{k=1..n} a(k) ~ 4*n^3/3. - Amiram Eldar, Dec 31 2024

A363031 a(n) = sigma(6n+1). Sum of the divisors of 6n+1, n >= 0.

Original entry on oeis.org

1, 8, 14, 20, 31, 32, 38, 44, 57, 72, 62, 68, 74, 80, 108, 112, 98, 104, 110, 144, 133, 128, 160, 140, 180, 152, 158, 164, 183, 248, 182, 216, 194, 200, 252, 212, 256, 224, 230, 288, 242, 280, 288, 304, 324, 272, 278, 284, 307, 360, 352, 308, 314, 360, 434, 332, 338, 400, 350, 432, 381, 368, 374, 380, 576, 432

Offset: 0

Omar E. Pol, May 18 2023

The sum of divisors function A000203 seems to behave with a certain periodicity of period 6.

Michael De Vlieger, Table of n, a(n) for n = 0..9999

Partial sums give A363161.

Cf. A000203, A016921, A224613.

Array[DivisorSigma[1, 6 # + 1] &, 66, 0] (* Michael De Vlieger, Aug 27 2023 *)

a(n) = sigma(6*n+1); \\ Michel Marcus, Aug 28 2023

from sympy import divisor_sigma
def A363031(n): return divisor_sigma(6*n+1) # Chai Wah Wu, Sep 07 2023

a(n) = A000203(6*n+1).

a(n) = A000203(A016921(n)).

A224614 Primes p such that q = 2p^3-1 and 2p*q^2-1 are both prime.

Original entry on oeis.org

181, 199, 4363, 4549, 14563, 15073, 15739, 27361, 27901, 33469, 34231, 37123, 46279, 48271, 48673, 54193, 56101, 64591, 64609, 65539, 65731, 70183, 70891, 75703, 75979, 77659, 77863, 80953, 94309, 112573, 114889, 115153, 117361, 118189, 135799, 144751

Offset: 1

Pierre CAMI, Apr 12 2013

nonn

When A224610(i) = 1 then prime(i) is in this sequence.

Subsequence of A177104. - R. J. Mathar, Apr 19 2013

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A224610, A224613.

[p: p in PrimesUpTo(180000) | IsPrime(q) and IsPrime(2*p*q^2-1) where q is 2*p^3-1 ]; // Bruno Berselli, Apr 19 2013

Reap[For[p = 2, p < 200000, p = NextPrime[p], If[PrimeQ[q = 2*p^3 - 1] && PrimeQ[r = 2*p*q^2 - 1], Sow[p]]]][[2, 1]] (* Jean-François Alcover, Apr 19 2013 *)
bpQ[n_]:=Module[{c=2n^3-1},AllTrue[{c,2n*c^2-1},PrimeQ]]; Select[ Prime[ Range[ 15000]],bpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 05 2015 *)

A346867 Sum of divisors of the numbers that have middle divisors.

Original entry on oeis.org

1, 3, 7, 12, 15, 13, 28, 24, 31, 39, 42, 60, 31, 56, 72, 63, 48, 91, 90, 96, 78, 124, 57, 93, 120, 120, 168, 104, 127, 144, 144, 195, 96, 186, 121, 224, 180, 234, 112, 252, 171, 156, 217, 210, 280, 216, 248, 182, 360, 133, 312, 255, 252, 336, 240, 336, 168, 403, 372, 234

Offset: 1

Omar E. Pol, Aug 18 2021

nonn

The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram the width is >= 1.
Also the width on the main diagonal equals the number of middle divisors.
So knowing this characteristic shape we can know if a number has middle divisors (or not) and the number of them just by looking at the diagram, even ignoring the concept of middle divisors.
Therefore we can see a geometric pattern of the distribution of the numbers with middle divisors in the stepped pyramid described in A245092.
For the definition of "width" see A249351.

			a(4) = 12 because the sum of divisors of the fourth number that has middle divisors (i.e., 6) is 1 + 2 + 3 + 6 = 12.
On the other hand we can see that in the main diagonal of every diagram the width is >= 1 as shown below.
Illustration of initial terms:
m(n) = A071562(n).
.
   n   m(n) a(n)   Diagram
.                  _ _   _   _   _ _     _     _ _   _   _       _
   1    1    1    |_| | | | | | | | |   | |   | | | | | | |     | |
   2    2    3    |_ _|_| | | | | | |   | |   | | | | | | |     | |
                   _ _|  _|_| | | | |   | |   | | | | | | |     | |
   3    4    7    |_ _ _|    _|_| | |   | |   | | | | | | |     | |
                   _ _ _|  _|  _ _|_|   | |   | | | | | | |     | |
   4    6   12    |_ _ _ _|  _| |  _ _ _| |   | | | | | | |     | |
                   _ _ _ _| |_ _|_|    _ _|   | | | | | | |     | |
   5    8   15    |_ _ _ _ _|  _|     |  _ _ _|_| | | | | |     | |
   6    9   13    |_ _ _ _ _| |      _|_| |  _ _ _|_| | | |     | |
                              |  _ _|    _| |    _ _ _|_| |     | |
                   _ _ _ _ _ _| |  _|  _|  _|   |  _ _ _ _|     | |
   7   12   28    |_ _ _ _ _ _ _| |_ _|  _|  _ _| |    _ _ _ _ _| |
                                  |  _ _|  _|    _|   |    _ _ _ _|
                   _ _ _ _ _ _ _ _| |     |     |  _ _|   |
   8   15   24    |_ _ _ _ _ _ _ _| |  _ _|  _ _|_|       |
   9   16   31    |_ _ _ _ _ _ _ _ _| |  _ _|  _|      _ _|
                   _ _ _ _ _ _ _ _ _| | |     |      _|
  10   18   39    |_ _ _ _ _ _ _ _ _ _| |  _ _|    _|
                   _ _ _ _ _ _ _ _ _ _| | |       |
  11   20   42    |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|
                                          | |
                                          | |
                   _ _ _ _ _ _ _ _ _ _ _ _| |
  12   24   60    |_ _ _ _ _ _ _ _ _ _ _ _ _|
.
The n-th diagram has the property that at least it shares a vertex with the (n+1)-st diagram.

Cf. A000203, A067742, A071090, A071561, A071562, A237591, A237593, A240542, A245092, A249351, A262626, A281007, A299777, A346864.

Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346865 (of hexagonal numbers), A346866 (of second hexagonal numbers), A346868 (of numbers with no middle divisors).

s[n_] := Module[{d = Divisors[n]}, If[AnyTrue[d, Sqrt[n/2] <= # < Sqrt[n*2] &], Plus @@ d, 0]]; Select[Array[s, 150], # > 0 &] (* Amiram Eldar, Aug 19 2021 *)

is(n) = fordiv(n, d, if(d^2>=n/2 && d^2<2*n, return(1))); 0 ; \\ A071562
apply(sigma, select(is, [1..200])) \\ Michel Marcus, Aug 19 2021

a(n) = A000203(A071562(n)).

A224626 Primes p such that q=2p^3-1, r=2p*q^2-1, and s=2pr^2-1 are all prime.

Original entry on oeis.org

27361, 65731, 167623, 424093, 1559449, 2389693, 3880633, 4683661, 5755921, 5780881, 6124411, 6840643, 7802959, 7822879, 7917769, 8876719, 9488683, 9640321, 9966139, 10392073, 10865083, 10988743, 12363991, 12457681, 12756253, 13471561, 14437561, 14508709, 14550331, 14839711, 15366223, 16574143

Offset: 1

Pierre CAMI, Apr 12 2013

nonn

A prime p here is prime p(n) when A224611(n) = 1.

A subsequence of A224614. - M. F. Hasler, Apr 22 2013

Pierre CAMI, Table of n, a(n) for n = 1..747

Cf. A224611, A224613, A224614.

Reap[ For[p = 2, p < 2*10^7, p = NextPrime[p], If[PrimeQ[q = 2*p^3 - 1] && PrimeQ[r = 2*p*q^2 - 1] && PrimeQ[2*p*r^2 - 1], Print[p]; Sow[p]] ]][[2, 1]] (* Jean-François Alcover, Apr 22 2013 *)
apQ[n_]:=Module[{q=2n^3-1,r},r=2n q^2-1;And@@PrimeQ[{q,r,2n r^2-1}]]; Select[ Prime[Range[1100000]],apQ] (* Harvey P. Dale, Nov 24 2013 *)

A319526 Square array read by antidiagonals upwards: T(n,k) = sigma(n*k), n >= 1, k >= 1.

Original entry on oeis.org

1, 3, 3, 4, 7, 4, 7, 12, 12, 7, 6, 15, 13, 15, 6, 12, 18, 28, 28, 18, 12, 8, 28, 24, 31, 24, 28, 8, 15, 24, 39, 42, 42, 39, 24, 15, 13, 31, 32, 60, 31, 60, 32, 31, 13, 18, 39, 60, 56, 72, 72, 56, 60, 39, 18, 12, 42, 40, 63, 48, 91, 48, 63, 40, 42, 12, 28, 36, 72, 91, 90, 96, 96, 90, 91, 72, 36, 28

Offset: 1

Omar E. Pol, Sep 25 2018

			The corner of the square array begins:
A000203:    1,   3,   4,   7,   6,  12,   8,  15,  13,  18,  12,  28, ...
A062731:    3,   7,  12,  15,  18,  28,  24,  31,  39,  42,  36,  60, ...
A144613:    4,  12,  13,  28,  24,  39,  32,  60,  40,  72,  48,  91, ...
A193553:    7,  15,  28,  31,  42,  60,  56,  63,  91,  90,  84, 124, ...
A283118:    6,  18,  24,  42,  31,  72,  48,  90,  78,  93,  72, 168, ...
A224613:   12,  28,  39,  60,  72,  91,  96, 124, 120, 168, 144, 195, ...
A283078:    8,  24,  32,  56,  48,  96,  57, 120, 104, 144,  96, 224, ...
A283122:   15,  31,  60,  63,  90, 124, 120, 127, 195, 186, 180, 252, ...
A283123:   13,  39,  40,  91,  78, 120, 104, 195, 121, 234, 156, 280, ...
...

Index entries for sequences related to sigma(n)

First 9 rows (also first 9 columns) are A000203, A062731, A144613, A193553, A283118, A224613, A283078, A283122, A283123.
Main diagonal gives A065764.
Cf. A000203, A003991, A216626, A319073.

Table[DivisorSigma[1, # k] &[m - k + 1], {m, 12}, {k, m}] // Flatten (* Michael De Vlieger, Dec 31 2018 *)

T(n,k) = A000203(n*k).

T(n,k) = A000203(A003991(n,k)).

A372675 a(n) = Sum_{j=1..n} Sum_{k=1..n} sigma(j*k).

Original entry on oeis.org

1, 14, 59, 190, 401, 914, 1499, 2632, 4113, 6424, 8645, 13284, 17023, 23092, 30715, 40484, 48711, 63890, 75351, 95792, 116421, 139822, 159911, 199176, 229499, 267438, 309283, 364462, 404933, 482792, 532553, 611208, 688593, 772540, 862471, 998760, 1083615, 1200328

Offset: 1

Vaclav Kotesovec, May 10 2024

nonn

Sum_{j=1..n} sigma(j*k) ~ A069097(k) * Pi^2 * n^2 / (12*k).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n)/n^4 for n = 1..100000

Cf. A069097, A372633, A372674.
Cf. A024916, A326124.
Cf. A000203, A062731, A144613, A193553, A283118, A224613, A283078, A283122, A283123.

Table[Sum[DivisorSigma[1, j*k], {j, 1, n}, {k, 1, n}], {n, 1, 50}]
s = 1; Join[{1}, Table[s += DivisorSigma[1, n^2] + 2*Sum[DivisorSigma[1, j*n], {j, 1, n - 1}], {n, 2, 50}]]

a(n) ~ c * n^4, where c = Pi^4 / (144*zeta(3)) = 0.56274...

A380580 Irregular tetrahedron T(s,r,k) read by rows in which the slice s is an irregular triangle, itself read by rows, in which row r lists the r-th row of A237593 sandwiched between two A380579(s+1,r+1), with s >= 0; 0 <= r <= s; k >= 0. Assume that row 0 of A237593 is empty.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 1, 1, 4, 4, 3, 1, 1, 3, 2, 2, 2, 2, 5, 5, 4, 1, 1, 4, 3, 2, 2, 3, 2, 2, 1, 1, 2, 2, 7, 7, 6, 1, 1, 6, 5, 2, 2, 5, 4, 2, 1, 1, 2, 4, 3, 3, 1, 1, 3, 3, 8, 8, 7, 1, 1, 7, 6, 2, 2, 6, 5, 2, 1, 1, 2, 5, 4, 3, 1, 1, 3, 4, 3, 3, 2, 2, 3, 3, 10, 10, 9, 1, 1, 9, 8, 2, 2, 8, 7, 2, 1, 1, 2, 7, 6, 3, 1, 1, 3, 6, 5, 3, 2, 2, 3, 5

Offset: 0

Omar E. Pol, Jan 27 2025

The discussion of this sequence was too long to be included here, and can be found in the attached "Discussion" text file (see the first link). - N. J. A. Sloane, Jul 31 2025

Paolo Xausa, Table of n, a(n) for n = 0..14441 (slices 0..50 of the tetrahedron, flattened).
Omar E. Pol, Discussion of the Sequence A380580.
Omar E. Pol, Illustration of initial terms of A000203 in the pyramid.
Omar E. Pol, Illustration of initial terms of A001065 in the pyramid.
Omar E. Pol, Illustration of initial terms of A048050 in the pyramid.
Omar E. Pol, Illustration of initial terms of A067742 in the pyramid.
Omar E. Pol, Illustration of initial terms of A224613 (black spiders).
Omar E. Pol, Illustration of initial terms of A237593 (essentially a template for the Pop-Up pyramid).
Omar E. Pol, Prism of partitions of 10 and its companion tower (both have the same volume).
Omar E. Pol, The symmetric representation of sigma(n), n = 1..64 (the top view of the pyramid and of the tower).

See the "Discussion" text file for the cross-references.

A237593row[n_] := Join[#, Reverse[#]] & [Table[Ceiling[(n+1)/k - (k+1)/2] + Quotient[k*(k+3) - 2*n, 2*(k+1)], {k, Quotient[Sqrt[8*n + 1] - 1, 2]}]];
A380580slice[s_] := Table[Join[#, A237593row[r], #] & [{Quotient[3*s, 2] - r + 1}], {r, 0, s}];
Array[A380580slice, 10, 0] (* Paolo Xausa, Aug 19 2025 *)

Edited by N. J. A. Sloane, Jul 31 2025

cp's OEIS Frontend

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