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Dirichlet g.f.: (zeta(s-3) / zeta(s))^2.
1

%I #8 Oct 15 2020 03:59:44

%S 1,14,52,161,248,728,684,1680,2080,3472,2660,8372,4392,9576,12896,

%T 16576,9824,29120,13716,39928,35568,37240,24332,87360,46376,61488,

%U 74412,110124,48776,180544,59580,157696,138320,137536,169632,334880,101304,192024,228384,416640

%N Dirichlet g.f.: (zeta(s-3) / zeta(s))^2.

%C Dirichlet convolution of Jordan function J_3 (A059376) with itself.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Jordan%27s_totient_function">Jordan's totient function</a>

%F Multiplicative with a(p^e) = p^(3*e - 6) * (p^6 + e * (p^3 - 1)^2 - 1).

%F a(n) = Sum_{d|n} J_3(d) * J_3(n/d).

%F a(n) = Sum_{d|n} d^3 * tau(d) * A007427(n/d), where tau = A000005.

%F (1/tau(n)) * Sum_{d|n} a(d) * tau(n/d) = n^3.

%F Sum_{k=1..n} a(k) ~ 2025 * n^4 * ((log(n) + 2*gamma - 1/4)/Pi^8 - 180*zeta'(4) / Pi^12), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Oct 14 2020

%t Jordan3[n_] := Sum[d^3 MoebiusMu[n/d], {d, Divisors[n]}]; a[n_] := Sum[Jordan3[d] Jordan3[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 40}]

%t a[1] = 1; f[p_, e_] := p^(3 e - 6) (p^6 + e (p^3 - 1)^2 - 1); a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 40}]

%Y Cf. A000005, A007427, A029935, A059376, A338164.

%K nonn,mult

%O 1,2

%A _Ilya Gutkovskiy_, Oct 14 2020