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A321322
a(n) = Sum_{d|n} mu(n/d)*J_2(d), where J_2() is the Jordan function (A007434).
4
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
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
KEYWORD
nonn,mult,easy
AUTHOR
Ilya Gutkovskiy, Nov 04 2018
STATUS
approved