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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Omar E. Pol, Aug 18 2021

Keywords

Comments

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.

Examples

			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    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
.

Crossrefs

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).

Programs

  • Mathematica
    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 *)
  • PARI
    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

Formula

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

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

Views

Author

Omar E. Pol, Aug 17 2021

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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).

Programs

  • Mathematica
    a[n_] := DivisorSigma[1, n*(2*n + 1)]; Array[a, 50] (* Amiram Eldar, Aug 18 2021 *)
  • PARI
    a(n) = sigma(n*(2*n + 1)); \\ Michel Marcus, Aug 18 2021

Formula

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

Views

Author

Omar E. Pol, Aug 18 2021

Keywords

Comments

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.

Examples

			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.

Crossrefs

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).

Programs

  • Mathematica
    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 *)
  • PARI
    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

Formula

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

A347108 a(n) = Sum_{k=1..n} sigma(k)*sigma(2*k), where sigma(n) = A000203(n) is the sum of the divisors of n.

Original entry on oeis.org

3, 24, 72, 177, 285, 621, 813, 1278, 1785, 2541, 2973, 4653, 5241, 6585, 8313, 10266, 11238, 14787, 15987, 19767, 22839, 25863, 27591, 35031, 37914, 42030, 46830, 53550, 56250, 68346, 71418, 79419, 86331, 93135, 100047, 117792, 122124, 130524, 139932, 156672
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 18 2021

Keywords

Crossrefs

Cf. A000203, A024916, A065764, A072379, A074285, A326123, A326124, A330322, A346865.

Programs

  • Mathematica
    Accumulate[Table[DivisorSigma[1,k] * DivisorSigma[1,2*k], {k, 1, 100}]]
  • PARI
    a(n) = sum(k=1, n, sigma(k)*sigma(2*k)); \\ Michel Marcus, Aug 18 2021

Formula

a(n) ~ 2*zeta(3)*n^3.

A347155 Sum of divisors of nontriangular numbers.

Original entry on oeis.org

3, 7, 6, 8, 15, 13, 12, 28, 14, 24, 31, 18, 39, 20, 42, 36, 24, 60, 31, 42, 40, 30, 72, 32, 63, 48, 54, 48, 38, 60, 56, 90, 42, 96, 44, 84, 72, 48, 124, 57, 93, 72, 98, 54, 120, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 68, 126, 96, 144, 72, 195, 74, 114, 124, 140
Offset: 1

Views

Author

Omar E. Pol, Aug 20 2021

Keywords

Comments

The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram both Dyck paths have the same orientation, that is both Dyck paths have peaks or both Dyck paths have valleys.
So knowing this characteristic shape we can know if a number is a nontriangular number (or not) just by looking at the diagram, even ignoring the concept of nontriangular number.
Therefore we can see a geometric pattern of the distribution of the nontriangular numbers in the stepped pyramid described in A245092.
If both Dyck paths have peaks on the main diagonal then the related subsequence of nontriangular numbers A014132 is A317303.
If both Dyck paths have valleys on the main diagonal then the related subsequence of nontriangular numbers A014132 is A317304.

Examples

			a(6) = 13 because the sum of divisors of the 6th nontriangular (i.e., 9) is 1 + 3 + 9 = 13.
On the other we can see that in the main diagonal of the diagrams both Dyck paths have the same orientation, that is both Dyck paths have peaks or both Dyck paths have valleys as shown below.
Illustration of initial terms:
m(n) = A014132(n).
.
   n   m(n) a(n)   Diagram
.                    _   _ _   _ _ _   _ _ _ _   _ _ _ _ _   _ _ _ _ _ _
                   _| | | | | | | | | | | | | | | | | | | | | | | | | | |
   1    2    3    |_ _|_| | | | | | | | | | | | | | | | | | | | | | | | |
                   _ _|  _|_| | | | | | | | | | | | | | | | | | | | | | |
   2    4    7    |_ _ _|    _|_| | | | | | | | | | | | | | | | | | | | |
   3    5    6    |_ _ _|  _|  _ _|_| | | | | | | | | | | | | | | | | | |
                   _ _ _ _|  _| |  _ _|_| | | | | | | | | | | | | | | | |
   4    7    8    |_ _ _ _| |_ _|_|    _ _|_| | | | | | | | | | | | | | |
   5    8   15    |_ _ _ _ _|  _|     |  _ _ _|_| | | | | | | | | | | | |
   6    9   13    |_ _ _ _ _| |      _|_| |  _ _ _|_| | | | | | | | | | |
                   _ _ _ _ _ _|  _ _|    _| |    _ _ _|_| | | | | | | | |
   7   11   12    |_ _ _ _ _ _| |  _|  _|  _|   |  _ _ _ _|_| | | | | | |
   8   12   28    |_ _ _ _ _ _ _| |_ _|  _|  _ _| | |  _ _ _ _|_| | | | |
   9   13   14    |_ _ _ _ _ _ _| |  _ _|  _|    _| | |    _ _ _ _|_| | |
  10   14   24    |_ _ _ _ _ _ _ _| |     |     |  _|_|   |  _ _ _ _ _|_|
                   _ _ _ _ _ _ _ _| |  _ _|  _ _|_|       | | |
  11   16   31    |_ _ _ _ _ _ _ _ _| |  _ _|  _|      _ _|_| |
  12   17   18    |_ _ _ _ _ _ _ _ _| | |_ _ _|      _| |  _ _|
  13   18   39    |_ _ _ _ _ _ _ _ _ _| |  _ _|    _|  _|_|
  14   19   20    |_ _ _ _ _ _ _ _ _ _| | |       |_ _|
  15   20   42    |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|  _|
                   _ _ _ _ _ _ _ _ _ _ _| | |  _ _| |
  16   22   36    |_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _|
  17   23   24    |_ _ _ _ _ _ _ _ _ _ _ _| | |
  18   24   60    |_ _ _ _ _ _ _ _ _ _ _ _ _| |
  19   25   31    |_ _ _ _ _ _ _ _ _ _ _ _ _| |
  20   26   42    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
  21   27   40    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Column m gives the nontriangular numbers.
Also the diagrams have on the main diagonal the following property: diagram [1] has peaks, diagrams [2, 3] have valleys, diagrams [4, 5, 6] have peaks, diagrams [7, 8, 9, 10] have valleys, and so on.
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 3 + 3 = 6, so a(3) = 6.
For more information see A237593.

Crossrefs

Cf. A000203, A000217, A014132, A237591, A237593, A245092, A262626, A317303, A317304, A346873.
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), A346868 (of numbers with no middle divisors).

Programs

Formula

a(n) = A000203(A014132(n)).
Showing 1-5 of 5 results.

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