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

1, 6, 16, 33, 48, 96, 96, 168, 208, 288, 240, 528, 336, 576, 768, 816, 576, 1248, 720, 1584, 1536, 1440, 1056, 2688, 1776, 2016, 2448, 3168, 1680, 4608, 1920, 3840, 3840, 3456, 4608, 6864, 2736, 4320, 5376, 8064, 3360, 9216, 3696, 7920, 9984, 6336, 4416, 13056, 7008, 10656

Offset: 1

Author

Ilya Gutkovskiy, Oct 14 2020

Keywords

• nonn
• mult

Comments

Dirichlet convolution of Jordan function J_2 (A007434) with itself.

Links

• Wikipedia, Jordan's totient function

Crossrefs

Cf. A000005, A007427, A007434, A029935, A034714, A306379, A321322, A338165.

Programs

Mathematica

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}]
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}]

Formula

Multiplicative with a(p^e) = p^(2*e - 4) * (p^4 + e * (p^2 - 1)^2 - 1).

a(n) = Sum_{d|n} J_2(d) * J_2(n/d).

a(n) = Sum_{d|n} d^2 * tau(d) * A007427(n/d), where tau = A000005.

a(n) = Sum_{d|n} d^2 * A321322(n/d).

(1/tau(n)) * Sum_{d|n} a(d) * tau(n/d) = n^2.

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