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

1,2

Comments

Möbius transform applied twice to squares.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

N. J. A. Sloane, Transforms.

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A020895 A174247 A343963 * A343495 A065139 A093916

Adjacent sequences: A321319 A321320 A321321 * A321323 A321324 A321325

Keyword

nonn,mult,easy

Author

Ilya Gutkovskiy, Nov 04 2018

Status

approved