A332653 a(n) = (1/n) * Sum_{k=1..n} n^(k/gcd(n, k)).

1, 2, 5, 19, 157, 1306, 19609, 266372, 5321721, 101001214, 2593742461, 61920391842, 1941507093541, 56984643437138, 2076518238897649, 72340172854919941, 3041324492229179281, 121440691499123469858, 5784852794328402307381, 262799364106291328009626

Offset: 1

Author

Ilya Gutkovskiy, Feb 18 2020

Keywords

• nonn

Crossrefs

Cf. A023037, A056665, A057661, A226561, A228640, A332620, A332621, A332652, A332655.

Programs

Magma

[(1/n)*&+[n^(k div Gcd(n,k)):k in [1..n]]:n in [1..21]]; // Marius A. Burtea, Feb 18 2020

Mathematica

Table[(1/n) Sum[n^(k/GCD[n, k]), {k, 1, n}], {n, 1, 20}]
Table[Sum[Sum[If[GCD[k, d] == 1, n^(k - 1), 0], {k, 1, d}], {d, Divisors[n]}], {n, 1, 20}]

Formula

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

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

a(n) = A332652(n) / n.