A332517 a(n) = Sum_{k=1..n} gcd(n,k)^n.

1, 5, 29, 274, 3129, 47515, 823549, 16843268, 387459861, 10009769725, 285311670621, 8918311856102, 302875106592265, 11112685048729175, 437893951473411261, 18447025557276459016, 827240261886336764193, 39346558373052524325225, 1978419655660313589123997

Offset: 1

Author

Ilya Gutkovskiy, Feb 14 2020

Keywords

• nonn

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

Crossrefs

Cf. A000010, A008683, A018804, A031971, A069097, A226561, A228640, A321294.

Programs

Magma

[&+[Gcd(n,k)^n:k in [1..n]]: n in [1..20]]; // Marius A. Burtea, Feb 15 2020

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}]

PARI

a(n) = sum(k=1, n, gcd(n, k)^n); \\ Michel Marcus, Feb 14 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

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)