A332621 a(n) = (1/n) * Sum_{k=1..n} n^(n/gcd(n, k)).

1, 3, 19, 133, 2501, 15631, 705895, 8389641, 258280489, 4000040011, 259374246011, 2972033984173, 279577021469773, 4762288684702095, 233543408203327951, 9223372037928525841, 778579070010669895697, 13115469358498302735067, 1874292305362402347591139

Offset: 1

Author

Ilya Gutkovskiy, Feb 17 2020

Keywords

• nonn

Links

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

Crossrefs

Cf. A000010, A056665, A130586, A226561, A228640, A308814, A321349, A332620.

Programs

Magma

[(1/n)*&+[n^(n div Gcd(n,k)):k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 17 2020

Mathematica

Table[(1/n) Sum[n^(n/GCD[n, k]), {k, 1, n}], {n, 1, 19}]
Table[(1/n) Sum[EulerPhi[d] n^d, {d, Divisors[n]}], {n, 1, 19}]
Table[SeriesCoefficient[Sum[Sum[EulerPhi[j] n^(j - 1) x^(k j), {j, 1, n}], {k, 1, n}], {x, 0, n}], {n, 1, 19}]

PARI

a(n) = sum(k=1, n, n^(n/gcd(n, k)))/n; \\ Michel Marcus, Mar 10 2021

Formula

a(n) = [x^n] Sum_{k>=1} Sum_{j>=1} phi(j) * n^(j-1) * x^(k*j).

a(n) = (1/n) * Sum_{k=1..n} n^(lcm(n, k)/k).

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

a(n) = A332620(n) / n.