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%I #8 Jan 13 2024 13:06:53
%S 1,-1,3,-2,3,-3,3,-2,6,-3,3,-6,3,-3,9,-1,3,-6,3,-6,9,-3,3,-6,6,-3,10,
%T -6,3,-9,3,1,9,-3,9,-12,3,-3,9,-6,3,-9,3,-6,18,-3,3,-3,6,-6,9,-6,3,
%U -10,9,-6,9,-3,3,-18,3,-3,18,4,9,-9,3,-6,9,-9,3,-12,3,-3,18
%N Dirichlet g.f.: zeta(s)^3 * (1 - 2^(1-s))^2.
%F Sum_{k=1..n} a(k) ~ n * log(2)^2.
%F Multiplicative with a(2^e) = (e^2-5*e+2)/2, and a(p^e) = (e+1)*(e+2)/2 for an odd prime p. - _Amiram Eldar_, Jan 13 2024
%t Table[Sum[Sum[-(-1)^d, {d, Divisors[k]}]*(-1)^(n/k+1), {k, Divisors[n]}], {n, 1, 100}]
%t f[p_, e_] := (e + 1)*(e + 2)/2; f[2, e_] := (e^2 - 5*e + 2)/2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 13 2024 *)
%o (PARI) a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e=f[i,2]; if(p == 2, (e^2-5*e+2)/2, (e+1)*(e+2)/2));} \\ _Amiram Eldar_, Jan 13 2024
%Y Cf. A007425, A048272, A288571.
%K sign,mult
%O 1,3
%A _Vaclav Kotesovec_, Jan 13 2024
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