A224613 -id:A224613 - cp's OEIS frontend

A384230 Number of subparts in the central part of the symmetric representation of sigma(n).

1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 0, 2, 1, 0, 3, 0, 2, 0, 0, 0, 2, 1, 0, 0, 2, 0, 4, 0, 1, 0, 0, 2, 3, 0, 0, 0, 2, 0, 4, 0, 0, 4, 0, 0, 2, 1, 1, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 2, 1, 0, 4, 0, 0, 0, 2, 0, 3, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 4, 0, 0, 0

Offset: 1

Omar E. Pol, Jun 29 2025

This sequence shares infinitely many terms with A067742 from which first differs at a(18). It also shares with A067742 the positions of zeros and nonzeros.
Observation: consider the 2-dense sublists of divisors of n. At least for the first 88 terms a(n) coincides with the number of odd terms in the central 2-dense sublist of divisors of n. For more information see A384225 and A280940.
See the "Discussion" text file in the first link for more comments.

			See the "Discussion" text file in the first link for the examples.

Omar E. Pol, Discussion of the Sequence A384230.
Omar E. Pol, Illustration of initial terms of A000203 in the pyramid.
Omar E. Pol, Illustration of initial terms of A001065 in the pyramid.
Omar E. Pol, Illustration of initial terms of A048050 in the pyramid.
Omar E. Pol, Illustration of initial terms of A067742 in the pyramid.
Omar E. Pol, Illustration of initial terms of A224613 (black spiders).
Omar E. Pol, Illustration of initial terms of A237593 (essentially a template for the Pop-Up pyramid).
Omar E. Pol, Prism of partitions of 10 and its companion tower (both have the same volume).
Omar E. Pol, The symmetric representation of sigma(n), n = 1..64 (the top view of the pyramid and of the tower).

Cf. A001227 (number of subparts), A071561 (positions of zeros), A071562 (positions of nonzeros), A237270 (parts), A237271, A237593, A279387 (subparts), A280940, A384225, A335574, A338488, A377654.

See the "Discussion" text file in the first link for more cross-references.

a(n) = 0 if and only if A067742(n) = 0.
a(n) >= A067742(n).
(a(n) - A067742(n)) is an even number.

Edited by Omar E. Pol, Aug 24 2025

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

Omar E. Pol, Aug 20 2021

nonn

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.

			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.

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).

Array[DivisorSigma[1,#+Round@Sqrt[2#]]&,100] (* Giorgos Kalogeropoulos, Aug 20 2021 *)

a(n) = sigma(n + round(sqrt(2*n))); \\ Michel Marcus, Aug 21 2021

a(n) = A000203(A014132(n)).

A365412 a(n) = sigma(6n+2). Sum of the divisors of 6n+2, n >= 0.

Original entry on oeis.org

3, 15, 24, 42, 42, 63, 60, 84, 93, 120, 96, 126, 114, 186, 132, 168, 171, 210, 216, 210, 186, 255, 204, 336, 222, 300, 240, 294, 324, 372, 336, 336, 294, 465, 312, 378, 330, 504, 432, 420, 399, 480, 384, 588, 480, 558, 420, 504, 540, 570, 456, 672, 474, 762, 492, 588, 549, 660, 744

Offset: 0

Omar E. Pol, Sep 07 2023

Consider a spiral similar to the spiral described in A239660 but instead of having four quadrants on the square grid the new spiral has six wedges on the triangular grid. A "diamond" formed by two adjacent triangles has area 1. a(n) is the number of diamonds (or the area) added in the second wedge after n turns. The interesting fact is that for n >> 1 the geometric pattern in the second wedge of the spiral is similar to the geometric pattern of the fourth wedge but it is different from the other wedges.

Other members of the same family are A363031 and A224613. Also 6*A098098.
Partial sums give A365442.
Cf. A000203, A016933, A239660.

Table[DivisorSigma[1, 6*n + 2], {n, 0, 60}] (* Amiram Eldar, Sep 09 2023 *)

a(n) = A000203(6*n+2).

a(n) = A000203(A016933(n)).

A365414 a(n) = sigma(6n+4). Sum of the divisors of 6n+4, n >= 0.

Original entry on oeis.org

7, 18, 31, 36, 56, 54, 90, 72, 98, 90, 127, 144, 140, 126, 180, 144, 217, 162, 248, 180, 224, 252, 270, 216, 266, 288, 378, 252, 308, 270, 360, 360, 399, 306, 434, 324, 504, 342, 450, 432, 434, 468, 511, 396, 476, 414, 720, 504, 518, 450, 620, 576, 560, 576, 630, 504, 756, 522, 756, 540

Offset: 0

Omar E. Pol, Sep 07 2023

Consider a spiral similar to the spiral described in A239660 but instead of having four quadrants on the square grid the new spiral has six wedges on the triangular grid. A "diamond" formed by two adjacent triangles has area 1. a(n) is the number of diamonds (or the area) added in the fourth wedge after n turns. The interesting fact is that for n >> 1 the geometric pattern in the fourth wedge of the spiral is similar to the geometric pattern of the second wedge but it is different from the other wedges.

Partial sums give A365444.
Other members of the same family are A363031 and A224613. Also 6*A098098.
Cf. A000203, A016957, A363031.

Table[DivisorSigma[1, 6*n + 4], {n, 0, 60}] (* Amiram Eldar, Sep 09 2023 *)

a(n) = sigma(6*n+4); \\ Michel Marcus, Sep 08 2023

a(n) = A000203(6*n+4).

a(n) = A000203(A016957(n)).

A224627 Prime numbers p such that 2p^3-1, 2p*q^2-1, 2pr^2-1, and 2ps^2-1 are prime numbers.

Original entry on oeis.org

19460899, 86276401, 87980803, 167646631, 300722029, 343507111, 479516311, 906597943, 998757829, 1031308249, 1112697199, 1311383431, 1962194053

Offset: 1

Pierre CAMI, Apr 12 2013

nonn

Subsequence of A224612, p = prime(n) when A224612(n)=1.

Cf. A224612, A224613, A224614, A224626.

Reap[ For[p = 2, p < 2*10^9, p = NextPrime[p], If[PrimeQ[q = 2*p^3 - 1] && PrimeQ[r = 2*p*q^2 - 1] && PrimeQ[s = 2*p*r^2 - 1] && PrimeQ[2*p*s^2 - 1], Print[p]; Sow[p]] ]][[2, 1]] (* Jean-François Alcover, Apr 22 2013 *)

More terms from Jean-François Alcover, Apr 22 2013

cp's OEIS Frontend

A384230 Number of subparts in the central part of the symmetric representation of sigma(n).

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A347155 Sum of divisors of nontriangular numbers.

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A365412 a(n) = sigma(6*n+2). Sum of the divisors of 6*n+2, n >= 0.

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A365414 a(n) = sigma(6*n+4). Sum of the divisors of 6*n+4, n >= 0.

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A224627 Prime numbers p such that 2*p^3-1, 2*p*q^2-1, 2*p*r^2-1, and 2*p*s^2-1 are prime numbers.

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A365412 a(n) = sigma(6n+2). Sum of the divisors of 6n+2, n >= 0.

A365414 a(n) = sigma(6n+4). Sum of the divisors of 6n+4, n >= 0.

A224627 Prime numbers p such that 2p^3-1, 2p*q^2-1, 2pr^2-1, and 2ps^2-1 are prime numbers.