A139256 -id:A139256 - cp's OEIS frontend

A262626 Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593.

1, 1, 1, 3, 2, 2, 2, 2, 2, 1, 1, 2, 7, 3, 1, 1, 3, 3, 3, 3, 2, 2, 3, 12, 4, 1, 1, 1, 1, 4, 4, 4, 4, 2, 1, 1, 2, 4, 15, 5, 2, 1, 1, 2, 5, 5, 3, 5, 5, 2, 2, 2, 2, 5, 9, 9, 6, 2, 1, 1, 1, 1, 2, 6, 6, 6, 6, 3, 1, 1, 1, 1, 3, 6, 28, 7, 2, 2, 1, 1, 2, 2, 7, 7, 7, 7, 3, 2, 1, 1, 2, 3, 7, 12, 12, 8, 3, 1, 2, 2, 1, 3, 8, 8, 8, 8, 8, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Sep 26 2015

Also the rows of both triangles A237270 and A237593 interleaved.
Also, irregular triangle read by rows in which T(n,k) is the area of the k-th region (from left to right in ascending diagonal) of the n-th symmetric set of regions (from the top to the bottom in descending diagonal) in the two-dimensional diagram of the perspective view of the infinite stepped pyramid described in A245092 (see the diagram in the Links section).
The diagram of the symmetric representation of sigma is also the top view of the pyramid, see Links section. For more information about the diagram see also A237593 and A237270.
The number of cubes at the n-th level is also A024916(n), the sum of all divisors of all positive integers <= n.
Note that this pyramid is also a quarter of the pyramid described in A244050. Both pyramids have infinitely many levels.
Odd-indexed rows are also the rows of the irregular triangle A237270.
Even-indexed rows are also the rows of the triangle A237593.
Lengths of the odd-indexed rows are in A237271.
Lengths of the even-indexed rows give 2*A003056.
Row sums of the odd-indexed rows gives A000203, the sum of divisors function.
Row sums of the even-indexed rows give the positive even numbers (see A005843).
Row sums give A245092.
From the front view of the stepped pyramid emerges a geometric pattern which is related to A001227, the number of odd divisors of the positive integers.
The connection with the odd divisors of the positive integers is as follows: A261697 --> A261699 --> A237048 --> A235791 --> A237591 --> A237593 --> A237270 --> this sequence.

			Irregular triangle begins:
  1;
  1, 1;
  3;
  2, 2;
  2, 2;
  2, 1, 1, 2;
  7;
  3, 1, 1, 3;
  3, 3;
  3, 2, 2, 3;
  12;
  4, 1, 1, 1, 1, 4;
  4, 4;
  4, 2, 1, 1, 2, 4;
  15;
  5, 2, 1, 1, 2, 5;
  5, 3, 5;
  5, 2, 2, 2, 2, 5;
  9, 9;
  6, 2, 1, 1, 1, 1, 2, 6;
  6, 6;
  6, 3, 1, 1, 1, 1, 3, 6;
  28;
  7, 2, 2, 1, 1, 2, 2, 7;
  7, 7;
  7, 3, 2, 1, 1, 2, 3, 7;
  12, 12;
  8, 3, 1, 2, 2, 1, 3, 8;
  8, 8, 8;
  8, 3, 2, 1, 1, 1, 1, 2, 3, 8;
  31;
  9, 3, 2, 1, 1, 1, 1, 2, 3, 9;
  ...
Illustration of the odd-indexed rows of triangle as the diagram of the symmetric representation of sigma which is also the top view of the stepped pyramid:
.
   n  A000203    A237270    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
   1     1   =      1      |_| | | | | | | | | | | | | | | |
   2     3   =      3      |_ _|_| | | | | | | | | | | | | |
   3     4   =    2 + 2    |_ _|  _|_| | | | | | | | | | | |
   4     7   =      7      |_ _ _|    _|_| | | | | | | | | |
   5     6   =    3 + 3    |_ _ _|  _|  _ _|_| | | | | | | |
   6    12   =     12      |_ _ _ _|  _| |  _ _|_| | | | | |
   7     8   =    4 + 4    |_ _ _ _| |_ _|_|    _ _|_| | | |
   8    15   =     15      |_ _ _ _ _|  _|     |  _ _ _|_| |
   9    13   =  5 + 3 + 5  |_ _ _ _ _| |      _|_| |  _ _ _|
  10    18   =    9 + 9    |_ _ _ _ _ _|  _ _|    _| |
  11    12   =    6 + 6    |_ _ _ _ _ _| |  _|  _|  _|
  12    28   =     28      |_ _ _ _ _ _ _| |_ _|  _|
  13    14   =    7 + 7    |_ _ _ _ _ _ _| |  _ _|
  14    24   =   12 + 12   |_ _ _ _ _ _ _ _| |
  15    24   =  8 + 8 + 8  |_ _ _ _ _ _ _ _| |
  16    31   =     31      |_ _ _ _ _ _ _ _ _|
  ...
The above diagram arises from a simpler diagram as shown below.
Illustration of the even-indexed rows of triangle as the diagram of the deployed front view of the corner of the stepped pyramid:
.
.                                 A237593
Level                               _ _
1                                 _|1|1|_
2                               _|2 _|_ 2|_
3                             _|2  |1|1|  2|_
4                           _|3   _|1|1|_   3|_
5                         _|3    |2 _|_ 2|    3|_
6                       _|4     _|1|1|1|1|_     4|_
7                     _|4      |2  |1|1|  2|      4|_
8                   _|5       _|2 _|1|1|_ 2|_       5|_
9                 _|5        |2  |2 _|_ 2|  2|        5|_
10              _|6         _|2  |1|1|1|1|  2|_         6|_
11            _|6          |3   _|1|1|1|1|_   3|          6|_
12          _|7           _|2  |2  |1|1|  2|  2|_           7|_
13        _|7            |3    |2 _|1|1|_ 2|    3|            7|_
14      _|8             _|3   _|1|2 _|_ 2|1|_   3|_             8|_
15    _|8              |3    |2  |1|1|1|1|  2|    3|              8|_
16   |9                |3    |2  |1|1|1|1|  2|    3|                9|
...
The number of horizontal line segments in the n-th level in each side of the diagram equals A001227(n), the number of odd divisors of n.
The number of horizontal line segments in the left side of the diagram plus the number of the horizontal line segment in the right side equals A054844(n).
The total number of vertical line segments in the n-th level of the diagram equals A131507(n).
The diagram represents the first 16 levels of the pyramid.
The diagram of the isosceles triangle and the diagram of the top view of the pyramid shows the connection between the partitions into consecutive parts and the sum of divisors function (see also A286000 and A286001). - _Omar E. Pol_, Aug 28 2018
The connection between the isosceles triangle and the stepped pyramid is due to the fact that this object can also be interpreted as a pop-up card. - _Omar E. Pol_, Nov 09 2022

Robert Price, Table of n, a(n) for n = 1..16048 (n=1..412 rows)
Omar E. Pol, An infinite stepped pyramid (A237593, A237270, A262626)
Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Perspective view of the stepped pyramid (levels: 1..16)
Index entries for sequences related to sigma(n)

Famous sequences that are visible in the stepped pyramid:
Cf. A000040 (prime numbers)......., for the characteristic shape see A346871.
Cf. A000079 (powers of 2)........., for the characteristic shape see A346872.
Cf. A000203 (sum of divisors)....., total area of the terraces in the n-th level.
Cf. A000217 (triangular numbers).., for the characteristic shape see A346873.
Cf. A000225 (Mersenne numbers)...., for a visualization see A346874.
Cf. A000384 (hexagonal numbers)..., for the characteristic shape see A346875.
Cf. A000396 (perfect numbers)....., for the characteristic shape see A346876.
Cf. A000668 (Mersenne primes)....., for a visualization see A346876.
Cf. A001097 (twin primes)........., for a visualization see A346871.
Cf. A001227 (# of odd divisors)..., number of subparts in the n-th level.
Cf. A002378 (oblong numbers)......, for a visualization see A346873.
Cf. A008586 (multiples of 4)......, perimeters of the successive levels.
Cf. A008588 (multiples of 6)......, for the characteristic shape see A224613.
Cf. A013661 (zeta(2))............., (area of the horizontal faces)/(n^2), n -> oo.
Cf. A014105 (second hexagonals)..., for the characteristic shape see A346864.
Cf. A067742 (# of middle divisors), # cells in the main diagonal in n-th level.
Other sequences that are visible in the stepped pyramid: A000096, A001065, A001359, A001747, A002939, A002943, A003056, A004125, A004277, A004526, A005279, A006512, A007606, A007607, A082647, A008438, A008578, A008864, A010814, A014106, A014107, A014132, A014574, A016945, A019434, A024206, A024916, A028552, A028982, A028983, A034856, A038550, A047836, A048050, A052928, A054735, A054844, A062731, A065091, A065475, A071561, A071562, A071904, A092506, A100484, A108605, A139256, A139257, A144396, A152677, A152678, A153485, A155085, A161680, A161983, A162917, A174905, A174973, A175254, A176810, A224880, A235791, A237270, A237271, A237591, A237593, A238005, A238524, A244049, A245092, A259176, A259177, A261348, A278972, A317302, A317303, A317304, A317305, A317307, A319529, A319796, A319801, A319802, A327329, A336305, (and several others).
Apart from zeta(2) other constants that are related to the stepped pyramid are A072691, A353908, A354238.
Cf. A054844, A131507, A196020, A236104, A237048, A239660, A244050, A259179, A261350, A261697, A261699, A262612, A280850, A286000, A286001, A296508.

A139257 Twice Mersenne primes A000668(n).

Original entry on oeis.org

6, 14, 62, 254, 16382, 262142, 1048574, 4294967294, 4611686018427387902, 1237940039285380274899124222, 324518553658426726783156020576254, 340282366920938463463374607431768211454

Offset: 1

Omar E. Pol, Apr 23 2008

nonn

Radicals of even perfect numbers. - Charles R Greathouse IV, Feb 01 2013

Amiram Eldar, Table of n, a(n) for n = 1..18
Florian Luca and Carl Pomerance, On the radical of a perfect number, New York Journal of Math., 16 (2010), 23-30; alternative link.

Cf. A000043, A000668, A000918, A060590, A095121, A100484, A139256.

2*(2^MersennePrimeExponent[Range[15]]-1) (* Harvey P. Dale, Jan 05 2020 *)

apply(p->2*(2^p-1),select(p->ispseudoprime(2^p-1),primes(40))) \\ Charles R Greathouse IV, Feb 01 2013

a(n) = 2*A000668(n).

a(n) = A000918(1 + A000043(n)) = A095121(A000043(n)). - Omar E. Pol, Jun 07 2012

Corrected and extended by Joerg Arndt, Jun 07 2012.

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)).

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

A139224 M(M-1)/2, where M is Mersenne prime A000668(n).

Original entry on oeis.org

3, 21, 465, 8001, 33542145, 8589737985, 137438167041, 2305843005992468481, 2658455991569831742348849606740148225, 191561942608236107294793377465333618488307184098607105

Offset: 1

Omar E. Pol, May 10 2008

nonn

Perfect number A000396(n) minus Mersenne prime A000668(n).

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000217, A000396, A000668, A036689, A139096, A139115, A139116, A139223, A139225, A139226, A139256, A139306.

a(n) = A000668(n)*(A000668(n)-1)/2.

a(n) = A000396(n)-A000668(n).

More terms from Max Alekseyev, Mar 09 2009

A324815 a(n) = 2*A156552(n) AND A323243(n), where AND is bitwise-and, A004198.

Original entry on oeis.org

0, 0, 0, 4, 0, 2, 0, 8, 12, 0, 0, 4, 0, 2, 16, 24, 0, 10, 0, 4, 36, 0, 0, 8, 24, 0, 24, 0, 0, 32, 0, 32, 4, 0, 40, 32, 0, 2, 128, 8, 0, 2, 0, 4, 36, 0, 0, 16, 48, 18, 4, 4, 0, 26, 72, 8, 512, 2, 0, 4, 0, 0, 12, 104, 8, 0, 0, 0, 4, 2, 0, 72, 0, 0, 32, 0, 80, 0, 0, 16, 8, 0, 0, 20, 256, 0, 2048, 0, 0, 74, 128, 0, 0, 0, 520, 56, 0, 32, 128, 64, 0, 2, 0, 8, 64

Offset: 1

Antti Karttunen, Mar 17 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A003987, A004198, A139256, A156552, A318468, A323243, A324201, A324716, A324816.

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552 by David A. Corneth
A324815(n) = bitand(2*A156552(n),A323243(n)); \\ Needs code also from A323243.

a(n) = 2*A156552(n) AND A323243(n), where AND is A004198.
a(n) = 2*A156552(n) - A324716(n) = 2*A156552(n) XOR A324716(n), where XOR is A003987.
For n > 1, a(n) = A318468(A156552(n)).
a(p) = 0 for all primes p.
a(A324201(n)) = A139256(n).
A000120(a(n)) = A324816(n).

A081756 Numbers n such that there is a proper divisor d of n satisfying sigma(d)=n.

Original entry on oeis.org

1, 12, 56, 360, 992, 2016, 16256, 120960, 131040, 1571328, 8714160, 67100672, 94279680, 182131200, 571963392, 1379454720, 4428914688, 5517818880, 17179738112, 70912195200, 153003540480, 159991977600, 175445913600, 265734881280, 274877382656, 612014161920

Offset: 1

Benoit Cloitre, Apr 08 2003

nonn

A139256 is a subsequence. - Michel Marcus, Dec 02 2013

Ray Chandler, Table of n, a(n) for n = 1..1000 (using comment in formula section from _David Wasserman_ and b-files for A007691 and A054030)

Cf. A007691, A054030, A065125, A139256.

kmax = 10^12;
A007691 = Cases[Import["https://oeis.org/A007691/b007691.txt", "Table"], {, }][[All, 2]];
A054030 = Cases[Import["https://oeis.org/A054030/b054030.txt", "Table"], {, }][[All, 2]];
okQ[n_] := AnyTrue[Most[Divisors[n]], DivisorSigma[1, #] == n&];
{1}~Join~Reap[Do[k = A007691[[i]]*A054030[[j]]; If[k <= kmax, Sow[k]], {i, Length[A007691]}, {j, Length[A054030]}]][[2, 1]] // Union // Select[#, okQ]& (* Jean-François Alcover, Oct 31 2019, after David Wasserman *)

Multiply A007691 by A054030 and sort the resulting sequence. - David Wasserman, Jun 28 2004

More terms from David Wasserman, Jun 28 2004

Description clarified by Ray Chandler, May 18 2017

A139223 M*(M-1), where M is Mersenne prime A000668(n).

Original entry on oeis.org

6, 42, 930, 16002, 67084290, 17179475970, 274876334082, 4611686011984936962, 5316911983139663484697699213480296450, 383123885216472214589586754930667236976614368197214210

Offset: 1

Omar E. Pol, May 10 2008

nonn

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000668, A036689, A139115, A139116, A139224, A139225, A139226, A139256.

a(n) = A000668(n)*(A000668(n)-1).

More terms from R. J. Mathar, Jun 24 2009

A139226 M(M-1)/6, where M is Mersenne prime A000668(n).

Original entry on oeis.org

1, 7, 155, 2667, 11180715, 2863245995, 45812722347, 768614335330822827, 886151997189943914116283202246716075, 63853980869412035764931125821777872829435728032869035, 4388012152856549445746584486819520216982214368341183846591146667, 4824670384888174809315457708695329493801178769338122219111555681805105015467

Offset: 1

Omar E. Pol, May 10 2008

nonn

Perfect number A000396(n) minus Mersenne prime A000668(n), divided by 3.

Terms from a(13) on have 313 or more digits and are not listed for that reason. - R. J. Mathar, May 11 2008

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000217, A000396, A000668, A036689, A139096, A139115, A139116, A139223, A139224, A139225, A139256, A139306.

a(n) = A000668(n)*(A000668(n)-1)/6 = A139223(n)/6 = A139224(n)/3.

a(n) = (A000396(n)-A000668(n))/3.

More terms from R. J. Mathar, May 11 2008

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

cp's OEIS Frontend

A262626 Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593.

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A139257 Twice Mersenne primes A000668(n).

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

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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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A139224 M(M-1)/2, where M is Mersenne prime A000668(n).

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A324815 a(n) = 2*A156552(n) AND A323243(n), where AND is bitwise-and, A004198.

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A081756 Numbers n such that there is a proper divisor d of n satisfying sigma(d)=n.

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A139223 M*(M-1), where M is Mersenne prime A000668(n).

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