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A368929 Dirichlet g.f.: zeta(s-2)^2 * (1 - 2^(3-s)) / zeta(s).

%I A368929 #25 Jan 12 2024 20:30:14
%S A368929 1,-1,17,-16,49,-17,97,-112,225,-49,241,-272,337,-97,833,-640,577,
%T A368929 -225,721,-784,1649,-241,1057,-1904,1825,-337,2673,-1552,1681,-833,
%U A368929 1921,-3328,4097,-577,4753,-3600,2737,-721,5729,-5488,3361,-1649,3697,-3856,11025,-1057,4417
%N A368929 Dirichlet g.f.: zeta(s-2)^2 * (1 - 2^(3-s)) / zeta(s).
%C A368929 Dirichlet convolution of A007434 and A162395.
%H A368929 Amiram Eldar, <a href="/A368929/b368929.txt">Table of n, a(n) for n = 1..10000</a>
%F A368929 Sum_{k=1..n} a(k) ~ log(2) * n^3 / (3*zeta(3)).
%F A368929 Multiplicative with a(2^e) = -(3*e-2)*2^(2*e-2), and a(p^e) = p^(2*e)*(1 + e*(1-1/p^2)) for an odd prime p. - _Amiram Eldar_, Jan 12 2024
%t A368929 Table[Sum[Sum[d^2 * MoebiusMu[k/d], {d, Divisors[k]}] * (-1)^(n/k + 1) * n^2/k^2, {k, Divisors[n]}], {n, 1, 100}]
%t A368929 f[p_, e_] := p^(2*e)*(1 + e*(1 - 1/p^2)); f[2, e_] := -(3*e - 2)*2^(2*e - 2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 12 2024 *)
%o A368929 (PARI) a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(p == 2, -(3*e-2)*2^(2*e-2), p^(2*e)*(1 + e*(1-1/p^2))));} \\ _Amiram Eldar_, Jan 12 2024
%Y A368929 Cf. A048272, A332794.
%Y A368929 Cf. A007434, A162395, A360428.
%K A368929 sign,mult
%O A368929 1,3
%A A368929 _Vaclav Kotesovec_, Jan 12 2024