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

Author

Ilya Gutkovskiy, Sep 08 2016

Keywords

• nonn
• easy

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).

Examples

			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}.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Proper divisors.

Crossrefs

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

Programs

Magma

[DivisorSigma(3, n) - n^3: n in [1..70]]; // Vincenzo Librandi, Sep 09 2016

Mathematica

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

PARI

a(n) = sigma(n, 3) - n^3; \\ Michel Marcus, Sep 08 2016

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