A321322 a(n) = Sum_{d|n} mu(n/d)*J_2(d), where J_2() is the Jordan function (A007434).

1, 2, 7, 9, 23, 14, 47, 36, 64, 46, 119, 63, 167, 94, 161, 144, 287, 128, 359, 207, 329, 238, 527, 252, 576, 334, 576, 423, 839, 322, 959, 576, 833, 574, 1081, 576, 1367, 718, 1169, 828, 1679, 658, 1847, 1071, 1472, 1054, 2207, 1008, 2304, 1152, 2009, 1503, 2807, 1152, 2737

Offset: 1

Author

Ilya Gutkovskiy, Nov 04 2018

Keywords

• nonn
• mult
• easy

Comments

Möbius transform applied twice to squares.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane, Transforms.

Crossrefs

Cf. A000010, A001615, A007427, A007431, A007433, A007434, A008683.

Programs

Mathematica

Table[Sum[MoebiusMu[n/d] Sum[MoebiusMu[d/j] j^2, {j, Divisors[d]}], {d, Divisors[n]}], {n, 55}]
nmax = 55; Rest[CoefficientList[Series[Sum[DivisorSum[k, MoebiusMu[#] MoebiusMu[k/#] &] x^k (1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]]
f[p_, e_] := If[e == 1, p^2 - 2, (p^2 - 1)^2*p^(2*e - 4)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 26 2020 *)

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^2/(1 - p^2*X))[n], ", ")) \\ Vaclav Kotesovec, Dec 11 2021

Formula

G.f.: Sum_{k>=1} A007427(k)*x^k*(1 + x^k)/(1 - x^k)^3.

a(n) = Sum_{d|n} mu(n/d)*phi(d)*psi(d), where phi() is the Euler totient function (A000010) and psi() is the Dedekind psi function (A001615).

Multiplicative with a(p^e) = p^2 - 2 if e = 1 and (p^2 - 1)^2 * p^(2*e - 4) otherwise. - Amiram Eldar, Oct 26 2020

From Vaclav Kotesovec, Dec 11 2021: (Start)

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

Sum_{k=1..n} a(k) ~ n^3 / (3*zeta(3)^2). (End)

a(n) = Sum_{1 <= i, j <= n} mu(gcd(i, j, n)). - Peter Bala, Jan 21 2024