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

%I A321322 #17 Jan 21 2024 15:28:57
%S A321322 1,2,7,9,23,14,47,36,64,46,119,63,167,94,161,144,287,128,359,207,329,
%T A321322 238,527,252,576,334,576,423,839,322,959,576,833,574,1081,576,1367,
%U A321322 718,1169,828,1679,658,1847,1071,1472,1054,2207,1008,2304,1152,2009,1503,2807,1152,2737
%N A321322 a(n) = Sum_{d|n} mu(n/d)*J_2(d), where J_2() is the Jordan function (A007434).
%C A321322 Möbius transform applied twice to squares.
%H A321322 Amiram Eldar, <a href="/A321322/b321322.txt">Table of n, a(n) for n = 1..10000</a>
%H A321322 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%F A321322 G.f.: Sum_{k>=1} A007427(k)*x^k*(1 + x^k)/(1 - x^k)^3.
%F A321322 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).
%F A321322 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
%F A321322 From _Vaclav Kotesovec_, Dec 11 2021: (Start)
%F A321322 Dirichlet g.f.: zeta(s-2) / zeta(s)^2.
%F A321322 Sum_{k=1..n} a(k) ~ n^3 / (3*zeta(3)^2). (End)
%F A321322 a(n) = Sum_{1 <= i, j <= n} mu(gcd(i, j, n)). - _Peter Bala_, Jan 21 2024
%t A321322 Table[Sum[MoebiusMu[n/d] Sum[MoebiusMu[d/j] j^2, {j, Divisors[d]}], {d, Divisors[n]}], {n, 55}]
%t A321322 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]]
%t A321322 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 *)
%o A321322 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - X)^2/(1 - p^2*X))[n], ", ")) \\ _Vaclav Kotesovec_, Dec 11 2021
%Y A321322 Cf. A000010, A001615, A007427, A007431, A007433, A007434, A008683.
%K A321322 nonn,mult,easy
%O A321322 1,2
%A A321322 _Ilya Gutkovskiy_, Nov 04 2018