A175926 -id:A175926 - cp's OEIS frontend

A275196 Odd numbers n such that sigma(n) does not divide sigma(n^3).

9, 25, 27, 49, 63, 75, 81, 99, 117, 121, 125, 135, 147, 153, 169, 171, 175, 207, 225, 243, 245, 261, 275, 289, 297, 325, 333, 343, 361, 363, 369, 375, 387, 405, 425, 441, 475, 477, 507, 513, 525, 529, 531, 539, 549, 567, 575, 603, 605, 625, 637, 639, 675, 693, 711, 725, 729

Offset: 1

Altug Alkan, Jul 20 2016

nonn

All terms are composite since sigma(p) = p + 1 and sigma(p^3) = p^3 + p^2 + p + 1 = (p + 1)(p^2 + 1) for p prime.
An odd number n with prime factorization Product_i p_i^(e_i) is in this sequence if and only if Product_i ((p_i^(3*e_i + 1) - 1)/(p_i^(e_i + 1) - 1)) is not an integer.
Nonsquare terms of this sequence are 27, 63, 75, 99, 117, 125, 135, 147, 153, 171, 175, 207, 243, 245, 261, 275, ...
Terms that are not perfect powers are 63, 75, 99, 117, 135, 147, 153, 171, 175, 207, 245, 261, 275, 297, 325, 333, 363, 369, 375, ...

			63 is a term because sigma(63^3) = 437200 is not divisible by sigma(63) = 104.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000203, A175926.

Select[2Range[400] - 1, Not[Divisible[DivisorSigma[1, #^3], DivisorSigma[1, #]]] &] (* Alonso del Arte, Jul 20 2016 *)

isok(n) = sigma(n^3) % sigma(n) != 0 && n % 2 == 1

A281364 Numbers n such that sigma(n^3) is the sum of two positive cubes.

Original entry on oeis.org

21, 22, 55, 129, 511, 770, 1070, 1071, 1074, 1434, 1708, 1914, 2721, 2926, 3080, 4195, 4464, 4814, 4879, 5236, 5907, 6086, 6114, 7228, 7831, 8029, 8289, 9086, 10149, 10547, 11145, 12305, 12621, 13348, 14993, 15012, 16212, 17670, 19513, 20020, 20083

Offset: 1

Altug Alkan, May 01 2016

nonn

265247 is the first term that is prime; sigma(265247^3) = 18661780598460480 = 48432^3 + 264708^3. In other words, the equation (1 + p)*(1 + p^2) = a^3 + b^3 where p is prime and a, b > 0, is soluble.

			21 is a term because sigma(21^3) = 16000 = 20^3 + 20^3.

Chai Wah Wu, Table of n, a(n) for n = 1..700

Cf. A000203, A000578, A003325, A175926.

isA003325(n) = for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1));
lista(nn) = for(n=1, nn, if(isA003325(sigma(n^3)), print1(n, ", ")));

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)).

A020476 Numbers k such that the sum of divisors of k^3 is a cube.

Original entry on oeis.org

1, 30154214043975990969, 76630991932799573643, 3590918978816938469301573291605, 68804810614900760202830061602802, 527113576787083306927808065765554478755087

Offset: 1

David W. Wilson

WARNING: the listed terms are not proved to be in order. According to Basso (2017), it is not even proved that a(2) = 30154214043975990969. - Max Alekseyev, Mar 25 2023

Andrea Basso, Power values of the sigma function, Bachelor’s Project Mathematics, University of Gronigen, July 2017. See Appendix D p. 36.

Cf. A000578, A175926.

isok(n) = ispower(sigma(n^3), 3); \\ Michel Marcus, Jul 19 2017

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A275196 Odd numbers n such that sigma(n) does not divide sigma(n^3).

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A281364 Numbers n such that sigma(n^3) is the sum of two positive cubes.

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A347155 Sum of divisors of nontriangular numbers.

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Formula

A020476 Numbers k such that the sum of divisors of k^3 is a cube.

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