A224613 -id:A224613 - cp's OEIS frontend

A283118 a(n) = sigma(5*n).

6, 18, 24, 42, 31, 72, 48, 90, 78, 93, 72, 168, 84, 144, 124, 186, 108, 234, 120, 217, 192, 216, 144, 360, 156, 252, 240, 336, 180, 372, 192, 378, 288, 324, 248, 546, 228, 360, 336, 465, 252, 576, 264, 504, 403, 432, 288, 744, 342, 468, 432, 588, 324, 720

Offset: 1

Seiichi Manyama, Mar 01 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sigma(k*n): A000203 (k=1), A062731 (k=2), A144613 (k=3), A193553 (k=4), this sequence (k=5), A224613 (k=6), A283078 (k=7), A283122 (k=8), A283123 (k=9).

Cf. A008587.

Table[DivisorSigma[1, 5 n], {n, 56}] (* Michael De Vlieger, Mar 01 2017 *)

a(n) = sigma(5*n) \\ Amiram Eldar, Dec 16 2022

a(n) = A000203(5*n).

Sum_{k=1..n} a(k) = (29*Pi^2/60) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

A372786 Number of divisors of 6n; a(n) = tau(6n) = A000005(6n).

Original entry on oeis.org

4, 6, 6, 8, 8, 9, 8, 10, 8, 12, 8, 12, 8, 12, 12, 12, 8, 12, 8, 16, 12, 12, 8, 15, 12, 12, 10, 16, 8, 18, 8, 14, 12, 12, 16, 16, 8, 12, 12, 20, 8, 18, 8, 16, 16, 12, 8, 18, 12, 18, 12, 16, 8, 15, 16, 20, 12, 12, 8, 24, 8, 12, 16, 16, 16, 18, 8, 16, 12, 24, 8, 20

Offset: 1

Vaclav Kotesovec, May 13 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A224613.

Cf. A000005, A001620, A099777, A372713, A372784, A372785, A372787, A372788, A372789, A372790, A372791, A372792.

Table[DivisorSigma[0, 6*n], {n, 1, 150}]

A372786(n) = numdiv(6*n); \\ Antti Karttunen, Jul 19 2024

Sum_{k=1..n} a(k) ~ (15*n*(log(n) + 2*gamma - 1) + n*(5*log(2) + 3*log(3))) / 6, where gamma is the Euler-Mascheroni constant A001620.

A346868 Sum of divisors of the numbers with no middle divisors.

Original entry on oeis.org

4, 6, 8, 18, 12, 14, 24, 18, 20, 32, 36, 24, 42, 40, 30, 32, 48, 54, 38, 60, 56, 42, 44, 84, 72, 48, 72, 98, 54, 72, 80, 90, 60, 62, 96, 84, 68, 126, 96, 72, 74, 114, 124, 140, 168, 80, 126, 84, 108, 132, 120, 90, 168, 128, 144, 120, 98, 102, 216, 104, 192, 162, 108, 110

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 equal to zero.
So knowing this characteristic shape we can know if a number has middle divisors  (or not) 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 no middle divisors in the stepped pyramid described in A245092.
For the definition of "width" see A249351.
All terms are even numbers.

			a(4) = 18 because the sum of divisors of the fourth number with no middle divisors (i.e., 10) is 1 + 2 + 5 + 10 = 18.
On the other hand we can see that in the main diagonal of every diagram the width is equal to zero as shown below.
Illustration of initial terms:
m(n) = A071561(n).
.
   n   m(n) a(n)   Diagram
.                      _   _   _     _ _   _ _     _   _   _ _ _     _
                      | | | | | |   | | | | | |   | | | | | | | |   | |
                   _ _|_| | | | |   | | | | | |   | | | | | | | |   | |
   1    3    4    |_ _|  _|_| | |   | | | | | |   | | | | | | | |   | |
                   _ _ _|    _|_|   | | | | | |   | | | | | | | |   | |
   2    5    6    |_ _ _|  _|    _ _| | | | | |   | | | | | | | |   | |
                   _ _ _ _|     |  _ _|_| | | |   | | | | | | | |   | |
   3    7    8    |_ _ _ _|  _ _|_|    _ _|_| |   | | | | | | | |   | |
                            |  _|     |  _ _ _|   | | | | | | | |   | |
                   _ _ _ _ _| |      _|_|    _ _ _|_| | | | | | |   | |
   4   10   18    |_ _ _ _ _ _|  _ _|       |    _ _ _|_| | | | |   | |
   5   11   12    |_ _ _ _ _ _| |  _|      _|   |  _ _ _ _|_| | |   | |
                   _ _ _ _ _ _ _| |      _|  _ _| | |  _ _ _ _|_|   | |
   6   13   14    |_ _ _ _ _ _ _| |  _ _|  _|    _| | |    _ _ _ _ _| |
   7   14   24    |_ _ _ _ _ _ _ _| |     |     |  _|_|   |  _ _ _ _ _|
                                    |  _ _|  _ _|_|       | |
                   _ _ _ _ _ _ _ _ _| |  _ _|  _|        _|_|
   8   17   18    |_ _ _ _ _ _ _ _ _| | |_ _ _|         |
                   _ _ _ _ _ _ _ _ _ _| |  _ _|        _|
   9   19   20    |_ _ _ _ _ _ _ _ _ _| | |        _ _|
                   _ _ _ _ _ _ _ _ _ _ _| |  _ _ _|
  10   21   32    |_ _ _ _ _ _ _ _ _ _ _| | |  _ _|
  11   22   36    |_ _ _ _ _ _ _ _ _ _ _ _| | |
  12   23   24    |_ _ _ _ _ _ _ _ _ _ _ _| | |
                                            | |
                   _ _ _ _ _ _ _ _ _ _ _ _ _| |
  13   26   42    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
.

Cf. A000203, A067742, A071090, A071561, A071562, A237591, A237593, 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), A346867 (of numbers with middle divisors).

s[n_] := Module[{d = Divisors[n]}, If[AnyTrue[d, Sqrt[n/2] <= # < Sqrt[n*2] &], 0, Plus @@ d]]; Select[Array[s, 110], # > 0 &] (* Amiram Eldar, Aug 19 2021 *)

is(n) = fordiv(n, d, if(sqrt(n/2) <= d && d < sqrt(2*n), return(0))); 1; \\ A071561 apply(sigma, select(is, [1..150])) \\ Michel Marcus, Aug 19 2021

a(n) = A000203(A071561(n)).

A346873 Triangle read by rows in which row n lists the row A000217(n) of A237591, n >= 1.

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 6, 2, 1, 1, 8, 3, 2, 1, 1, 11, 4, 3, 1, 1, 1, 15, 5, 3, 2, 1, 1, 1, 19, 6, 4, 2, 2, 1, 1, 1, 23, 8, 5, 2, 2, 2, 1, 1, 1, 28, 10, 5, 3, 3, 2, 1, 1, 1, 1, 34, 11, 6, 4, 3, 2, 2, 1, 1, 1, 1, 40, 13, 7, 5, 3, 2, 2, 2, 1, 1, 1, 1, 46, 16, 8, 5, 4, 2, 3

Offset: 1

Omar E. Pol, Aug 06 2021

The characteristic shape of the symmetric representation of sigma(A000217(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, or vice versa, the smallest Dyck path has a peak and the largest Dyck path has valley.
So knowing this characteristic shape we can know if a number is a triangular number (or not) just by looking at the diagram, even ignoring the concept of triangular number.
Therefore we can see a geometric pattern of the distribution of the triangular 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(A000217(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000217(n).
T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th triangular number into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th triangular number into exactly k + 1 consecutive parts.

			Triangle begins:
   1;
   2,  1;
   4,  1, 1;
   6,  2, 1, 1;
   8,  3, 2, 1, 1;
  11,  4, 3, 1, 1, 1;
  15,  5, 3, 2, 1, 1, 1;
  19,  6, 4, 2, 2, 1, 1, 1;
  23,  8, 5, 2, 2, 2, 1, 1, 1;
  28, 10, 5, 3, 3, 2, 1, 1, 1, 1;
  34, 11, 6, 4, 3, 2, 2, 1, 1, 1, 1;
  40, 13, 7, 5, 3, 2, 2, 2, 1, 1, 1, 1;
  46, 16, 8, 5, 4, 2, 3, 1, 2, 1, 1, 1, 1;
...
Illustration of initial terms:
Column T gives the triangular numbers (A000217).
Column S gives A074285, the sum of the divisors of the triangular numbers which equals the area (and the number of cells) of the associated diagram.
-------------------------------------------------------------------------
  n    T    S   Diagram
-------------------------------------------------------------------------
                 _   _     _       _         _           _             _
  1    1    1   |_| | |   | |     | |       | |         | |           | |
               1 _ _|_|   | |     | |       | |         | |           | |
  2    3    4   |_ _|  _ _| |     | |       | |         | |           | |
                  2  1|    _|     | |       | |         | |           | |
                 _ _ _|  _|    _ _| |       | |         | |           | |
  3    6   12   |_ _ _ _| 1   |  _ _|       | |         | |           | |
                    4    1 _ _|_|           | |         | |           | |
                          |  _|1       _ _ _|_|         | |           | |
                 _ _ _ _ _| | 1    _ _| |               | |           | |
  4   10   18   |_ _ _ _ _ _|2    |    _|               | |           | |
                      6          _|  _|          _ _ _ _|_|           | |
                                |_ _|1 1        | |                   | |
                                | 2            _| |                   | |
                 _ _ _ _ _ _ _ _|4            |  _|          _ _ _ _ _| |
  3   15   24   |_ _ _ _ _ _ _ _|          _ _|_|           |  _ _ _ _ _|
                        8              _ _|  _|1            | |
                                      |_ _ _|1 1         _ _| |
                                      |  3           _ _|  _ _|
                                      |4            |    _|
                 _ _ _ _ _ _ _ _ _ _ _|            _|  _|
  4   21   32   |_ _ _ _ _ _ _ _ _ _ _|      _ _ _|  _|1 1
                          11                |  _ _ _|2
                                            | |  3
                                            | |
                                            | |5
                 _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  5   28   56   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
                              15
.

Row sums give A000217, n >= 1.
Column 1 gives A039823.
For the characteristic shape of sigma(A000040(n)) see A346871.
For the characteristic shape of sigma(A000079(n)) see A346872.
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.
Cf. A000203, A074285, A237591, A237593, A245092, A249351, A262626.

T(n,k) = A237591(A000217(n),k). - Omar E. Pol, Feb 06 2023

Name corrected by Omar E. Pol, Feb 06 2023

A346876 Irregular triangle read by rows in which row n is the "n-th even perfect number" row of A237591, n >= 1.

Original entry on oeis.org

4, 1, 1, 15, 5, 3, 2, 1, 1, 1, 249, 83, 42, 25, 17, 13, 9, 7, 6, 5, 5, 3, 4, 2, 3, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4065, 1355, 678, 407, 271, 194, 146, 113, 91, 75, 62, 52, 45, 40, 34, 30, 27, 25, 22, 19, 19, 16, 15, 14, 13, 12, 12, 10, 10, 9, 9, 8, 8, 7

Offset: 1

Omar E. Pol, Aug 06 2021

The characteristic shape of the symmetric representation of sigma(A000396(n)) consists in that the diagram has only one region (or part) and that region has whidth 1 except in the main diagonal where the width is 2.
So knowing this characteristic shape we can know if a number is an even perfect number (or not) just by looking at the diagram, even ignoring the concept of even perfect number (see the examples).
Therefore we can see a geometric pattern of the distribution of the even perfect numbers in the stepped pyramid described in A245092.
For the definition of "width" see A249351.
T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A000396(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000396(n) assuming there are no odd perfect numbers.
T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th even perfect number into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th perfect number into exactly k + 1 consecutive parts.

			Triangle begins:
    4, 1, 1;
   15, 5, 3, 2, 1, 1,1;
  249,83,42,25,17,13,9,7,6,5,5,3,4,2,3,2,2,2,2,2,1,2,1,2,1,1,1,1,1,1,1;
...
Illustration of initial terms:
Column P gives the even perfect numbers (A000396 assuming there are no odd perfect numbers).
Column S gives A139256, the sum of the divisors of the even perfect numbers equals the area (and the number of cells) of the associated diagram.
-------------------------------------------------------------------------
  n    P   S    Diagram:   1                                           2
-------------------------------------------------------------------------
                           _                                           _
                          | |                                         | |
                          | |                                         | |
                       _ _| |                                         | |
                      |    _|                                         | |
                 _ _ _|  _|                                           | |
  1    6   12   |_ _ _ _| 1                                           | |
                    4    1                                            | |
                                                                      | |
                                                                      | |
                                                                      | |
                                                                      | |
                                                                      | |
                                                                      | |
                                                             _ _ _ _ _| |
                                                            |  _ _ _ _ _|
                                                            | |
                                                         _ _| |
                                                     _ _|  _ _|
                                                    |    _|
                                                   _|  _|
                                                  |  _|1 1
                                             _ _ _| | 1
                                            |  _ _ _|2
                                            | |  3
                                            | |
                                            | |5
                 _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  2   28   56   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
                              15
.
For n = 3, P = 496, the diagram is too large to include here. To draw that diagram note that the lengths of the line segments of the smallest Dyck path are [248, 83, 42, 25, 17, 13, 9, 7, 6, 5, 5, 3, 4, 2, 3, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 4, 3, 5, 5, 6, 7, 9, 13, 17, 25, 42, 83, 248] and the lengths of the line segments of the largest Dyck path are [249, 83, 42, 25, 17, 13, 9, 7, 6, 5, 5, 3, 4, 2, 3, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 4, 3, 5, 5, 6, 7, 9, 13, 17, 25, 42, 83, 249].

Michel Marcus, Table of n, a(n) for n = 1..8359 (rows 1..5).

Row sums give A000396.
Row lengths give A000668.
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(A008588(n)) see A224613.
Cf. A000203, A139256, A237591, A237593, A245092, A249351, A262626.

row235791(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i);
row(n) = {my(orow = concat(row235791(n), 0)); vector(#orow -1, i, orow[i] - orow[i+1]); } \\ A237591
tabf(nn) = {for (n=1, nn, my(p=prime(n)); if (isprime(2^n-1), print(row(2^(n-1)*(2^n-1)));););}
tabf(7) \\ Michel Marcus, Aug 31 2021

More terms from Michel Marcus, Aug 31 2021

Name edited by Michel Marcus, Jun 16 2023

A346871 Irregular triangle read by rows in which row n lists the row A000040(n) of A237591, n >= 1.

Original entry on oeis.org

2, 2, 1, 3, 2, 4, 2, 1, 6, 3, 1, 1, 7, 3, 2, 1, 9, 4, 2, 1, 1, 10, 4, 2, 2, 1, 12, 5, 2, 2, 1, 1, 15, 6, 3, 2, 1, 1, 1, 16, 6, 3, 2, 2, 1, 1, 19, 7, 4, 2, 2, 1, 1, 1, 21, 8, 4, 2, 2, 2, 1, 1, 22, 8, 4, 3, 2, 1, 2, 1, 24, 9, 4, 3, 2, 2, 1, 1, 1, 27, 10, 5, 3, 2, 2, 1, 2, 1

Offset: 1

Omar E. Pol, Aug 06 2021

The characteristic shape of the symmetric representation of sigma(prime(n)) consists in that the diagram contains exactly two regions (or parts) and each region is a rectangle (or bar), except for the first prime number (the 2) whose symmetric representation of sigma(2) consists of only one region which contains three cells.
So knowing this characteristic shape we can know if a number is prime (or not) just by looking at the diagram, even ignoring the concept of prime number.
Therefore we can see a geometric pattern of the exact distribution of prime numbers in the stepped pyramid described in A245092.
T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(prime(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000040(n).
T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th prime into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th prime into exactly k + 1 consecutive parts.

			Triangle begins:
   2;
   2, 1;
   3, 2;
   4, 2, 1;
   6, 3, 1, 1;
   7, 3, 2, 1;
   9, 4, 2, 1, 1;
  10, 4, 2, 2, 1;
  12, 5, 2, 2, 1, 1;
  15, 6, 3, 2, 1, 1, 1;
  16, 6, 3, 2, 2, 1, 1;
  19, 7, 4, 2, 2, 1, 1, 1;
  21, 8, 4, 2, 2, 2, 1, 1;
  22, 8, 4, 3, 2, 1, 2, 1;
  24, 9, 4, 3, 2, 2, 1, 1, 1;
...
Illustration of initial terms:
Row 1:    _
        _| |
       |_ _|
         2                         Semilength = 2
.
Row 2:      _
           | |
        _ _|_|
       |_ _|1                      Semilength = 3
         2
.
Row 3:          _
               | |
               | |
              _|_|
        _ _ _|                     Semilength = 5
       |_ _ _|2
          3
.
Row 4:              _
                   | |
                   | |
                   | |
                  _|_|
                _|
        _ _ _ _| 1                 Semilength = 7
       |_ _ _ _|2
           4
.
Row 5:                         _
                              | |
                              | |
                              | |
                              | |
                              | |
                           _ _|_|
                         _|
                       _|1         Semilength = 11
                      |1
           _ _ _ _ _ _|
          |_ _ _ _ _ _|3
                6
.
The area (also the number of cells) of the successive diagrams gives A008864.

Row sums give A000040.
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.
Cf. A000203, A008864, A018252, A237591, A237593, A245092, A249351, A262626.

A346872 Irregular triangle read by rows in which row n lists the row 2^(n-1) of A237591, n >= 1.

Original entry on oeis.org

1, 2, 3, 1, 5, 2, 1, 9, 3, 2, 1, 1, 17, 6, 3, 2, 2, 1, 1, 33, 11, 6, 4, 2, 2, 2, 1, 2, 1, 65, 22, 11, 7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1, 129, 43, 22, 13, 9, 7, 5, 4, 3, 3, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 257, 86, 43, 26, 18, 12, 10, 8, 6, 5, 4, 4, 3, 3, 3, 2, 3

Offset: 1

Omar E. Pol, Aug 06 2021

The characteristic shape of the symmetric representation of sigma(2^(n-1)) consists in that the diagram contains exactly one region (or part) and that region has width 1.
So knowing this characteristic shape we can know if a number is power of 2 or not just by looking at the diagram, even ignoring the concept of power of 2.
Therefore we can see a geometric pattern of the distribution of the powers of 2 in the stepped pyramid described in A245092.
For the definition of "width" see A249351.
T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(2^(n-1)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000079(n-1).
T(n,k) is also the difference between the total number of partitions of all positive integers <= 2^(n-1) into exactly k consecutive parts, and the total number of partitions of all positive integers <= 2^(n-1) into exactly k + 1 consecutive parts.

			Triangle begins:
    1;
    2;
    3,  1;
    5,  2,  1;
    9,  3,  2,  1, 1;
   17,  6,  3,  2, 2, 1, 1;
   33, 11,  6,  4, 2, 2, 2, 1, 2, 1;
   65, 22, 11,  7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1;
  129, 43, 22, 13, 9, 7, 5, 4, 3, 3, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1;
...
Illustration of initial terms:
.
Row 1:  _
       |_|                              Semilength = 1
        1
Row 2:    _
        _| |
       |_ _|
         2                              Semilength = 2
.
Row 3:        _
             | |
            _| |
        _ _|  _|
       |_ _ _|1                         Semilength = 4
          3
.
Row 4:                _
                     | |
                     | |
                     | |
                  _ _| |
                _|  _ _|
               |  _|
        _ _ _ _| | 1                    Semilength = 8
       |_ _ _ _ _|2
            5
.
Row 5:                                _
                                     | |
                                     | |
                                     | |
                                     | |
                                     | |
                                     | |
                                     | |
                                _ _ _| |
                               |  _ _ _|
                              _| |
                            _|  _|
                        _ _|  _|        Semilength = 16
                       |  _ _|1 1
                       | | 2
        _ _ _ _ _ _ _ _| |3
       |_ _ _ _ _ _ _ _ _|
                9
.
The area (also the number of cells) of the successive diagrams gives the nonzero Mersenne numbers A000225.

Row sums give A000079.
Column 1 gives A094373.
For the characteristic shape of sigma(A000040(n)) see A346871.
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.
Cf. A000203, A000225, A237591, A237593, A245092, A249351, A262626.

A346874 Irregular triangle read by rows in which row n lists the row 2^n - 1 of A237591, n >= 1.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 3, 2, 1, 1, 16, 6, 3, 2, 2, 1, 1, 32, 11, 6, 4, 2, 2, 2, 1, 2, 1, 64, 22, 11, 7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1, 128, 43, 22, 13, 9, 7, 5, 4, 3, 3, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 256, 86, 43, 26, 18, 12, 10, 8, 6, 5, 4, 4, 3, 3, 3, 2, 3

Offset: 1

Omar E. Pol, Aug 06 2021

The Mersenne number A000225(n) does not has a characteristic shape of its symmetric representation of sigma(A000225(n)). On the other hand, we can find that number in two ways in the symmetric representation of the powers of 2 as follows: the Mersenne numbers are the semilength of the smallest Dyck path and also they equals the area (or the number of cells) of the region of the diagram (see examples).
Therefore we can see a geometric pattern of the distribution of the Mersenne numbers in the stepped pyramid described in A245092.
T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A000225(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000225(n).
T(n,k) is also the difference between the total number of partitions of all positive integers <= Mersenne number A000225(n) into k consecutive parts, and the total number of partitions of all positive integers <= Mersenne number A000225(n) into k + 1 consecutive parts.

			Triangle begins:
    1;
    2,  1;
    4,  2,  1;
    8,  3,  2,  1, 1;
   16,  6,  3,  2, 2, 1, 1;
   32, 11,  6,  4, 2, 2, 2, 1, 2, 1;
   64, 22, 11,  7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1;
  128, 43, 22, 13, 9, 7, 5, 4, 3, 3, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1;
...
Illustration of initial terms:
.
Row 1:
       0_                               Semilength = 0    Area = 1
       |_|
Row 2:
          _
       1_| |                            Semilength = 1    Area = 3
       |_ _|
.
Row 3:        _
             | |
         1  _| |
       2_ _|  _|                        Semilength = 3    Area = 7
       |_ _ _|
.
Row 4:                _
                     | |
                     | |
                     | |
                  _ _| |
              1 _|  _ _|
          4   2|  _|                    Semilength = 7    Area = 15
        _ _ _ _| |
       |_ _ _ _ _|
.
Row 5:                                _
                                     | |
                                     | |
                                     | |
                                     | |
                                     | |
                                     | |
                                     | |
                                _ _ _| |
                               |  _ _ _|
                              _| |
                         1 1_|  _|
                      2 _ _|  _|        Semilength = 15   Area = 31
                       |  _ _|
               8      3| |
        _ _ _ _ _ _ _ _| |
       |_ _ _ _ _ _ _ _ _|
.

Row sums give A000225, n >= 1.
Column 1 gives A000079.
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 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.
Cf. A000203, A237591, A237593, A245092, A249351, A262626.

A283119 Expansion of exp( Sum_{n>=1} sigma(6n)x^n/n ) in powers of x.

Original entry on oeis.org

1, 12, 86, 469, 2141, 8594, 31247, 104945, 330094, 982284, 2786861, 7584060, 19893185, 50494558, 124437410, 298555264, 699017259, 1600364304, 3589048673, 7896510620, 17067607791, 36283650153, 75947406513, 156672628539, 318804641925, 640390347979

Offset: 0

Seiichi Manyama, Mar 01 2017

nonn

sigma(6*n) = A000203(6*n), the sum of divisors of 6*n (A224613).

			G.f.: A(x) = 1 + 12*x + 86*x^2 + 469*x^3 + 2141*x^4 + 8594*x^5 + ...
log(A(x)) = 12*x + 28*x^2/2 + 39*x^3/3 + 60*x^4/4 + 72*x^5/5 + 91*x^6/6 + 96*x^7/7 + 124*x^8/8 + ... + sigma(6*n)*x^n/n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A224613 (sigma(6*n)), A283164 (exp( Sum_{n>=1} -sigma(6*n)*x^n/n )).

Cf. A182818 (k=2), A182819 (k=3), A182820 (k=4), A182821 (k=5), this sequence (k=6), A283077 (k=7), A283120 (k=8), A283121 (k=9).

Table[SeriesCoefficient[Product[(1 - x^(2 i))^4*(1 - x^(3 i))^3/((1 - x^i)^12*(1 - x^(6 i))), {i, n}], {x, 0, n}], {n, 0, 25}] (* Michael De Vlieger, Mar 01 2017 *)

G.f.: Product_{n>=1} (1 - x^(2*n))^4 * (1 - x^(3*n))^3/((1 - x^n)^12 * (1 - x^(6*n))).
a(n) = (1/n)*Sum_{k=1..n} sigma(6*k)*a(n-k). - Seiichi Manyama, Mar 05 2017
a(n) ~ 55^(7/4) * exp(sqrt(55*n)*Pi/3) / (41472*sqrt(3)*n^(9/4)). - Vaclav Kotesovec, Mar 20 2017

A283164 Expansion of exp( Sum_{n>=1} -sigma(6n)x^n/n ) in powers of x.

Original entry on oeis.org

1, -12, 58, -133, 95, 194, -418, 97, 325, -99, -238, 169, -217, 131, 190, -145, 441, -647, 169, -527, 72, 1129, 313, -972, 2, -491, -565, 1944, -1175, -216, 972, 863, -1259, 288, 0, -1155, -1355, -207, 2925, 1753, 1402, -2387, -2257, -1030, 315, 432, -72, 1621, 358

Offset: 0

Seiichi Manyama, Mar 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A224613 (sigma(6*n)), A283119 (exp( Sum_{n>=1} sigma(6*n)*x^n/n )).

Cf. exp( Sum_{n>=1} -sigma(k*n)*x^n/n ): A115110 (k=2), A185654 (k=3), A283163 (k=4), A282937 (k=5), this sequence (k=6), A282942 (k=7), A283168 (k=8), A283169 (k=9).

G.f.: Product_{n>=1} (1 - x^n)^12 * (1 - x^(6*n))/((1 - x^(2*n))^4 * (1 - x^(3*n))^3).

a(n) = -(1/n)*Sum_{k=1..n} sigma(6*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

cp's OEIS Frontend

A283118 a(n) = sigma(5*n).

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A372786 Number of divisors of 6n; a(n) = tau(6*n) = A000005(6*n).

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A346868 Sum of divisors of the numbers with no middle divisors.

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A346873 Triangle read by rows in which row n lists the row A000217(n) of A237591, n >= 1.

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A346876 Irregular triangle read by rows in which row n is the "n-th even perfect number" row of A237591, n >= 1.

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A346871 Irregular triangle read by rows in which row n lists the row A000040(n) of A237591, n >= 1.

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A346872 Irregular triangle read by rows in which row n lists the row 2^(n-1) of A237591, n >= 1.

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A346874 Irregular triangle read by rows in which row n lists the row 2^n - 1 of A237591, n >= 1.

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A283119 Expansion of exp( Sum_{n>=1} sigma(6*n)*x^n/n ) in powers of x.

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A372786 Number of divisors of 6n; a(n) = tau(6n) = A000005(6n).

A283119 Expansion of exp( Sum_{n>=1} sigma(6n)x^n/n ) in powers of x.

A283164 Expansion of exp( Sum_{n>=1} -sigma(6n)x^n/n ) in powers of x.