A346864 -id:A346864 - 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.

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)).

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

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.

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)).

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula