OFFSET
1,2
COMMENTS
If n is prime, a(n) = n-1 + n^n. - Robert Israel, Feb 16 2020
LINKS
Robert Israel, Table of n, a(n) for n = 1..386
FORMULA
a(n) = Sum_{d|n} phi(n/d) * d^n.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_(n-1)(d).
a(n) ~ n^n.
From Richard L. Ollerton, May 09 2021: (Start)
a(n) = Sum_{k=1..n} (n/gcd(n,k))^n*phi(gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))*gcd(n,k)*sigma_(n-1)(gcd(n,k))/phi(n/gcd(n,k)). (End)
MAPLE
f:= n -> add(igcd(n, k)^n, k=1..n):
map(f, [$1..30]); # Robert Israel, Feb 16 2020
MATHEMATICA
Table[Sum[GCD[n, k]^n, {k, 1, n}], {n, 1, 19}]
Table[Sum[EulerPhi[n/d] d^n, {d, Divisors[n]}], {n, 1, 19}]
Table[Sum[MoebiusMu[n/d] d DivisorSigma[n - 1, d], {d, Divisors[n]}], {n, 1, 19}]
PROG
(PARI) a(n) = sum(k=1, n, gcd(n, k)^n); \\ Michel Marcus, Feb 14 2020
(Magma) [&+[Gcd(n, k)^n:k in [1..n]]: n in [1..20]]; // Marius A. Burtea, Feb 15 2020
(Python)
from sympy import totient, divisors
def A332517(n):
return sum(totient(d)*(n//d)**n for d in divisors(n, generator=True)) # Chai Wah Wu, Feb 15 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 14 2020
STATUS
approved