A276634 Sum of cubes of proper divisors of n.

0, 1, 1, 9, 1, 36, 1, 73, 28, 134, 1, 316, 1, 352, 153, 585, 1, 981, 1, 1198, 371, 1340, 1, 2556, 126, 2206, 757, 3160, 1, 4752, 1, 4681, 1359, 4922, 469, 8605, 1, 6868, 2225, 9710, 1, 12600, 1, 12052, 4257, 12176, 1, 20476, 344, 16759, 4941, 19846, 1, 26496, 1457, 25624, 6887, 24398, 1

Offset

1,4

Comments

More generally, the Dirichlet generating function for the sum of k-th powers of proper divisors of n is zeta(s-k)*(zeta(s) - 1).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Proper divisors.

Formula

a(n) = 1 if n is prime.

a(p^k) = (p^(3*k) - 1)/(p^3 - 1) for p prime.

Dirichlet g.f.: zeta(s-3)*(zeta(s) - 1).

a(n) = A001158(n) - A000578(n).

A000035(a(n)) = A053867(n).

Sum_{n=1..k} a(n) ~ k^2*(Pi^4*k^2/90 - (k + 1)^2)/4.

G.f.: -x*(1 + 4*x + x^2)/(1 - x)^4 + Sum_{k>=1} k^3*x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 17 2017

Example

a(10) = 1^3 + 2^3 + 5^3 = 134, because 10 has 3 proper divisors {1,2,5}.
a(11) = 1^3 = 1, because 11 has 1 proper divisor {1}.

Mathematica

Table[DivisorSigma[3, n] - n^3, {n, 70}]

Prog

(PARI) a(n) = sigma(n, 3) - n^3; \\ Michel Marcus, Sep 08 2016
(Magma) [DivisorSigma(3, n) - n^3: n in [1..70]]; // Vincenzo Librandi, Sep 09 2016

Crossrefs

Cf. A000035, A000578, A001065, A001158, A032741, A053867, A067558.

Sequence in context: A195278 A092477 A019433 * A050312 A295575 A107892

Adjacent sequences: A276631 A276632 A276633 * A276635 A276636 A276637

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Sep 08 2016

Status

approved