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A332617 a(n) = Sum_{k=1..n} J_n(k), where J is the Jordan function, J_n(k) = k^n * Product_{p|k, p prime} (1 - 1/p^n).

%I A332617 #8 Sep 08 2022 08:46:25
%S A332617 1,4,34,336,4390,66312,1197858,24612000,574002448,14903406552,
%T A332617 427622607366,13419501812640,457579466056498,16840326075104280,
%U A332617 665473192580864556,28101209228393371200,1262896789586657015796,60182268296582518426368,3031282541337682050032664
%N A332617 a(n) = Sum_{k=1..n} J_n(k), where J is the Jordan function, J_n(k) = k^n * Product_{p|k, p prime} (1 - 1/p^n).
%H A332617 Wikipedia, <a href="http://en.wikipedia.org/wiki/Jordan%27s_totient_function">Jordan's totient function</a>
%F A332617 a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} Sum_{j>=1} mu(k) * j^n * x^(k*j).
%t A332617 Table[Sum[Sum[MoebiusMu[k/d] d^n, {d, Divisors[k]}], {k, 1, n}], {n, 1, 19}]
%t A332617 Table[SeriesCoefficient[(1/(1 - x)) Sum[Sum[MoebiusMu[k] j^n x^(k j), {j, 1, n}], {k, 1, n}], {x, 0, n}], {n, 1, 19}]
%o A332617 (Magma) [&+[&+[MoebiusMu(k div d)*d^n:d in Divisors(k)]:k in [1..n]]:n in [1..20]]; // _Marius A. Burtea_, Feb 17 2020
%Y A332617 Cf. A000010, A002088, A007434, A059376, A059377, A059378, A059379, A059380, A067858, A319194, A321879.
%K A332617 nonn
%O A332617 1,2
%A A332617 _Ilya Gutkovskiy_, Feb 17 2020