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A338164 Dirichlet g.f.: (zeta(s-2) / zeta(s))^2.

%I A338164 #9 Oct 15 2020 03:59:37
%S A338164 1,6,16,33,48,96,96,168,208,288,240,528,336,576,768,816,576,1248,720,
%T A338164 1584,1536,1440,1056,2688,1776,2016,2448,3168,1680,4608,1920,3840,
%U A338164 3840,3456,4608,6864,2736,4320,5376,8064,3360,9216,3696,7920,9984,6336,4416,13056,7008,10656
%N A338164 Dirichlet g.f.: (zeta(s-2) / zeta(s))^2.
%C A338164 Dirichlet convolution of Jordan function J_2 (A007434) with itself.
%H A338164 Wikipedia, <a href="http://en.wikipedia.org/wiki/Jordan%27s_totient_function">Jordan's totient function</a>
%F A338164 Multiplicative with a(p^e) = p^(2*e - 4) * (p^4 + e * (p^2 - 1)^2 - 1).
%F A338164 a(n) = Sum_{d|n} J_2(d) * J_2(n/d).
%F A338164 a(n) = Sum_{d|n} d^2 * tau(d) * A007427(n/d), where tau = A000005.
%F A338164 a(n) = Sum_{d|n} d^2 * A321322(n/d).
%F A338164 (1/tau(n)) * Sum_{d|n} a(d) * tau(n/d) = n^2.
%F A338164 Sum_{k=1..n} a(k) ~ ((3*log(n) + 6*gamma - 1)/(9*zeta(3)^2) - 2*zeta'(3) / (3*zeta(3)^3)) * n^3, where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Oct 14 2020
%t A338164 Jordan2[n_] := Sum[d^2 MoebiusMu[n/d], {d, Divisors[n]}]; a[n_] := Sum[Jordan2[d] Jordan2[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 50}]
%t A338164 a[1] = 1; f[p_, e_] := p^(2 e - 4) (p^4 + e (p^2 - 1)^2 - 1); a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 50}]
%Y A338164 Cf. A000005, A007427, A007434, A029935, A034714, A306379, A321322, A338165.
%K A338164 nonn,mult
%O A338164 1,2
%A A338164 _Ilya Gutkovskiy_, Oct 14 2020