OFFSET
1,4
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^(4*k) - 1)/(p^4 - 1) for p is prime.
Dirichlet g.f.: zeta(s-4)*(zeta(s) - 1).
G.f.: -x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5 + Sum_{k>=1} k^4 x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 18 2017
Sum_{k=1..n} a(k) ~ (Zeta(5) - 1)*n^5 / 5. - Vaclav Kotesovec, Feb 02 2019
EXAMPLE
a(10) = 1^4 + 2^4 + 5^4 = 642, because 10 has 3 proper divisors {1,2,5}.
a(11) = 1^4 = 1, because 11 has 1 proper divisor {1}.
MATHEMATICA
Table[DivisorSigma[4, n] - n^4, {n, 60}]
PROG
(PARI) for(n=1, 60, print1(sigma(n, 4) - n^4, ", ")) \\ Indranil Ghosh, Mar 18 2017
(Python)
from sympy.ntheory import divisor_sigma
print([divisor_sigma(n, 4) - n**4 for n in range(1, 61)]) # Indranil Ghosh, Mar 18 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Dec 10 2016
STATUS
approved