A342612 a(n) = Sum_{d|n} phi(n/d)^(n-d).

1, 2, 5, 10, 257, 50, 46657, 16450, 1679681, 327682, 10000000001, 4196098, 8916100448257, 15237476354, 4398063289345, 35184640528386, 18446744073709551617, 19747769389058, 39346408075296537575425

Offset: 1

Author

Seiichi Manyama, Mar 16 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..388

Crossrefs

Cf. A000010, A342489, A342619, A342629.

Programs

Mathematica

a[n_] := DivisorSum[n, EulerPhi[n/#]^(n - #) &]; Array[a, 20] (* Amiram Eldar, Mar 17 2021 *)

PARI

a(n) = sumdiv(n, d, eulerphi(n/d)^(n-d));

PARI

a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^(n-gcd(k, n)-1));

PARI

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)^(k-1)*x^k/(1-eulerphi(k)^(k-1)*x^k)))

Formula

a(n) = Sum_{k=1..n} phi(n/gcd(k,n))^(n - gcd(k,n) - 1).

G.f.: Sum_{k>=1} phi(k)^(k-1) * x^k/(1 - phi(k)^(k-1) * x^k).

If p is prime, a(p) = 1 + (p-1)^(p-1).