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A326828 a(n) = (1/2) * Sum_{d|n} mu(n/d) * phi(d) * (psi(d) + 1), where mu = A008683, phi = A000010 and psi = A001615.

%I A326828 #12 Dec 11 2021 04:02:32
%S A326828 1,1,4,5,13,7,26,19,34,23,64,32,89,47,82,74,151,64,188,105,167,119,
%T A326828 274,127,296,167,294,214,433,161,494,292,421,287,548,290,701,359,590,
%U A326828 417,859,329,944,540,742,527,1126,506,1170,576,1012,757,1429,576,1382
%N A326828 a(n) = (1/2) * Sum_{d|n} mu(n/d) * phi(d) * (psi(d) + 1), where mu = A008683, phi = A000010 and psi = A001615.
%C A326828 Moebius transform applied twice to triangular numbers (A000217).
%F A326828 G.f.: Sum_{i>=1} Sum_{j>=1} mu(j) * mu(i) * x^(i*j) / (1 - x^(i*j))^3.
%F A326828 Dirichlet g.f.: (zeta(s-1) + zeta(s-2)) / (2 * zeta(s)^2).
%F A326828 a(n) = (1/2) * Sum_{d|n} mu(n/d) * (phi(d) + J_2(d)), where J_2 = A007434.
%F A326828 a(n) = (1/2) * Sum_{d|n} d * (d + 1) * A007427(n/d).
%F A326828 a(n) = Sum_{d|n} mu(n/d) * A007438(d).
%F A326828 Sum_{k=1..n} a(k) ~ n^3 / (6*zeta(3)^2). - _Vaclav Kotesovec_, Dec 11 2021
%p A326828 with(numtheory):
%p A326828 b:= proc(n) option remember;
%p A326828        add(mobius(n/d)*d*(d+1)/2, d=divisors(n))
%p A326828     end:
%p A326828 a:= proc(n) option remember;
%p A326828        add(mobius(n/d)*b(d), d=divisors(n))
%p A326828     end:
%p A326828 seq(a(n), n=1..60);  # _Alois P. Heinz_, Oct 20 2019
%t A326828 Table[1/2 Sum[MoebiusMu[n/d] EulerPhi[d] (DirichletConvolve[j, MoebiusMu[j]^2, j, d] + 1), {d, Divisors[n]}], {n, 1, 55}]
%t A326828 Table[1/2 Sum[d (d + 1) DivisorSum[n/d, MoebiusMu[#] MoebiusMu[(n/d)/#] &], {d, Divisors[n]}], {n, 1, 55}]
%t A326828 nmax = 55; CoefficientList[Series[Sum[Sum[MoebiusMu[j] MoebiusMu[i] x^(i j)/(1 - x^(i j))^3, {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x] // Rest
%Y A326828 Cf. A000010, A000217, A001615, A007427, A007431, A007438, A008683, A321322, A326826.
%K A326828 nonn
%O A326828 1,3
%A A326828 _Ilya Gutkovskiy_, Oct 20 2019