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

%I A332621 #12 Sep 08 2022 08:46:25
%S A332621 1,3,19,133,2501,15631,705895,8389641,258280489,4000040011,
%T A332621 259374246011,2972033984173,279577021469773,4762288684702095,
%U A332621 233543408203327951,9223372037928525841,778579070010669895697,13115469358498302735067,1874292305362402347591139
%N A332621 a(n) = (1/n) * Sum_{k=1..n} n^(n/gcd(n, k)).
%H A332621 Seiichi Manyama, <a href="/A332621/b332621.txt">Table of n, a(n) for n = 1..386</a>
%F A332621 a(n) = [x^n] Sum_{k>=1} Sum_{j>=1} phi(j) * n^(j-1) * x^(k*j).
%F A332621 a(n) = (1/n) * Sum_{k=1..n} n^(lcm(n, k)/k).
%F A332621 a(n) = (1/n) * Sum_{d|n} phi(d) * n^d.
%F A332621 a(n) = A332620(n) / n.
%t A332621 Table[(1/n) Sum[n^(n/GCD[n, k]), {k, 1, n}], {n, 1, 19}]
%t A332621 Table[(1/n) Sum[EulerPhi[d] n^d, {d, Divisors[n]}], {n, 1, 19}]
%t A332621 Table[SeriesCoefficient[Sum[Sum[EulerPhi[j] n^(j - 1) x^(k j), {j, 1, n}], {k, 1, n}], {x, 0, n}], {n, 1, 19}]
%o A332621 (Magma) [(1/n)*&+[n^(n div Gcd(n,k)):k in [1..n]]:n in [1..20]]; // _Marius A. Burtea_, Feb 17 2020
%o A332621 (PARI) a(n) = sum(k=1, n, n^(n/gcd(n, k)))/n; \\ _Michel Marcus_, Mar 10 2021
%Y A332621 Cf. A000010, A056665, A130586, A226561, A228640, A308814, A321349, A332620.
%K A332621 nonn
%O A332621 1,2
%A A332621 _Ilya Gutkovskiy_, Feb 17 2020