A342471 a(n) = Sum_{d|n} phi(d)^n.

1, 2, 9, 18, 1025, 130, 279937, 65794, 10078209, 2097154, 100000000001, 16789506, 106993205379073, 156728328194, 35185445863425, 281479271743490, 295147905179352825857, 203119913861122, 708235345355337676357633, 1152923703631151106

Offset: 1

Author

Seiichi Manyama, Mar 13 2021

Keywords

• nonn

Crossrefs

Cf. A000010, A029939, A110567, A309369, A338997, A342470.

Programs

Mathematica

a[n_] := DivisorSum[n, EulerPhi[#]^n &]; Array[a, 20] (* Amiram Eldar, Mar 13 2021 *)

PARI

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

PARI

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

PARI

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (eulerphi(k)*x)^k/(1-(eulerphi(k)*x)^k)))

Formula

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

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

If p is prime, a(p) = 1 + (p-1)^p = A110567(p-1).

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