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

1,2

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

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

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 *)

Prog

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

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.

Sequence in context: A005337 A160891 A223425 * A038991 A068020 A131991

Adjacent sequences: A175923 A175924 A175925 * A175927 A175928 A175929

Keyword

nonn,mult

Author

Zak Seidov, Oct 19 2010

Status

approved