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).
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
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Sep 08 2016
STATUS
approved