A338165 Dirichlet g.f.: (zeta(s-3) / zeta(s))^2.

1, 14, 52, 161, 248, 728, 684, 1680, 2080, 3472, 2660, 8372, 4392, 9576, 12896, 16576, 9824, 29120, 13716, 39928, 35568, 37240, 24332, 87360, 46376, 61488, 74412, 110124, 48776, 180544, 59580, 157696, 138320, 137536, 169632, 334880, 101304, 192024, 228384, 416640

Offset: 1

Author

Ilya Gutkovskiy, Oct 14 2020

Keywords

• nonn
• mult

Comments

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

Links

• Wikipedia, Jordan's totient function

Crossrefs

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

Programs

Mathematica

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}]
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}]

Formula

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

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

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

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

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