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a(n) = Sum_{k=1..n} n^(n/gcd(n, k)).
5

%I #13 Sep 08 2022 08:46:25

%S 1,6,57,532,12505,93786,4941265,67117128,2324524401,40000400110,

%T 2853116706121,35664407810076,3634501279107049,66672041585829330,

%U 3503151123049919265,147573952606856413456,13235844190181388226849,236078448452969449231206,35611553801885644604231641

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

%H Seiichi Manyama, <a href="/A332620/b332620.txt">Table of n, a(n) for n = 1..385</a>

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

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

%F a(n) = Sum_{d|n} phi(d) * n^d.

%F a(n) = n * A332621(n).

%t Table[Sum[n^(n/GCD[n, k]), {k, 1, n}], {n, 1, 19}]

%t Table[Sum[EulerPhi[d] n^d, {d, Divisors[n]}], {n, 1, 19}]

%t Table[SeriesCoefficient[Sum[Sum[EulerPhi[j] n^j x^(k j), {j, 1, n}], {k, 1, n}], {x, 0, n}], {n, 1, 19}]

%o (Magma) [&+[n^(n div Gcd(n,k)):k in [1..n]]:n in [1..20]]; // _Marius A. Burtea_, Feb 17 2020

%o (PARI) a(n) = sum(k=1, n, n^(n/gcd(n, k))); \\ _Michel Marcus_, Mar 10 2021

%Y Cf. A000010, A056665, A066108, A226561, A228640, A321349, A332621.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Feb 17 2020