A175926 Sum of divisors of cubes.

1, 15, 40, 127, 156, 600, 400, 1023, 1093, 2340, 1464, 5080, 2380, 6000, 6240, 8191, 5220, 16395, 7240, 19812, 16000, 21960, 12720, 40920, 19531, 35700, 29524, 50800, 25260, 93600, 30784, 65535, 58560, 78300, 62400, 138811, 52060, 108600, 95200

Offset: 1

Author

Zak Seidov, Oct 19 2010

Keywords

• nonn
• mult

Comments

The Mobius transform of the sequence is 1, 14, 39 ,112, 155,..., which equals the sequence defined by n*A160889(n). - R. J. Mathar, Apr 15 2011

Zhi-Wei Sun noted that the first 10^7 terms are pairwise distinct, but Noam D. Elkies found that a(48142241) = a(48374911), a(384422506) = a(403764207) and so on. - Zhi-Wei Sun, Jan 08 2014

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. sigma(n^k): A000203 (k=1), A065764 (k=2), this sequence (k=3), A202994 (k=4), A203556 (k=5).

Cf. A000578, A013662, A160889.

Programs

Magma

[ SumOfDivisors(n^3) : n in [1..100]]; // Vincenzo Librandi, Apr 14 2011

Mathematica

DivisorSigma[1,#]&/@((Range[40])^3) (* Harvey P. Dale, Aug 30 2015 *)
f[p_, e_] := (p^(3*e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 10 2020 *)

PARI

a(n) = sigma(n^3); \\ Amiram Eldar, Nov 05 2022

Python

from math import prod
from sympy import factorint
def A175926(n): return prod((p**(3*e+1)-1)//(p-1) for p,e in factorint(n).items()) # Chai Wah Wu, Oct 25 2023

Formula

a(n) = A000203(n^3). - R. J. Mathar, Mar 31 2011

Multiplicative with a(p^e) = (p^(3e+1)-1)/(p-1). - R. J. Mathar, Mar 31 2011

Sum_{k>=1} 1/a(k) = 1.11535899887110289127674868460900333554265894187008102863022551119560512446... - Vaclav Kotesovec, Sep 20 2020

Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(4)/4) * Product_{p prime} (1 + 1/p^2 + 1/p^3) = 0.4732277044... . - Amiram Eldar, Nov 05 2022