A321349 a(n) = Sum_{d|n} phi(d^n), where phi() is the Euler totient function (A000010).

1, 3, 19, 137, 2501, 16071, 705895, 8421505, 258293449, 4007813013, 259374246011, 2972767821815, 279577021469773, 4762869973595499, 233543432626753439, 9223512776490647553, 778579070010669895697, 13115569455375954492093, 1874292305362402347591139

Offset: 1

Author

Ilya Gutkovskiy, Nov 06 2018

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..386

Crossrefs

Cf. A000010, A057660, A068963, A068970, A226459, A226561.

Programs

Mathematica

Table[Sum[EulerPhi[d^n], {d, Divisors[n]}], {n, 19}]
nmax = 19; Rest[CoefficientList[Series[Sum[k^(k - 1) EulerPhi[k] x^k/(1 - (k x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[(n/GCD[n, k])^(n - 1), {k, n}], {n, 19}]

PARI

a(n) = sumdiv(n, d, eulerphi(d^n)); \\ Michel Marcus, Nov 06 2018

Formula

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

a(n) = Sum_{d|n} d^(n-1)*phi(d).

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

From Richard L. Ollerton, May 08 2021: (Start)

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

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