A228640 a(n) = Sum_{d|n} phi(d)*n^(n/d).

0, 1, 6, 33, 280, 3145, 46956, 823585, 16781472, 387422001, 10000100440, 285311670721, 8916103479504, 302875106592409, 11112006930972780, 437893890382391745, 18446744078004651136, 827240261886336764449, 39346408075494964903956, 1978419655660313589124321

Offset: 0

Author

Alois P. Heinz, Aug 28 2013

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Main diagonal of A054618, A054619, A185651.

Cf. A000010, A056665.

Programs

Magma

[0] cat [&+[EulerPhi(d)*n^(n div d): d in Divisors(n)]:n in [1..20]]; // Marius A. Burtea, Feb 15 2020

Maple

with(numtheory):
a:= n-> add(phi(d)*n^(n/d), d=divisors(n)):
seq(a(n), n=0..20);

Mathematica

a[0] = 0; a[n_] := DivisorSum[n, EulerPhi[#]*n^(n/#)&]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 21 2017 *)

PARI

a(n) = if (n, sumdiv(n, d, eulerphi(d)*n^(n/d)), 0); \\ Michel Marcus, Feb 15 2020; corrected Jun 13 2022

PARI

a(n) = sum(k=1, n, n^gcd(k, n)); \\ Seiichi Manyama, Mar 10 2021

Python

from sympy import totient, divisors
def A228640(n):
    return sum(totient(d)*n**(n//d) for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

Formula

a(n) = Sum_{k=1..n} n^gcd(k,n) = n * A056665(n). - Seiichi Manyama, Mar 10 2021

a(n) = Sum_{k=1..n} n^(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021