cp's OEIS Frontend

A326415 Dirichlet g.f.: zeta(2*s) / zeta(s)^3.

%I A326415 #22 Feb 07 2023 15:11:57
%S A326415 1,-3,-3,4,-3,9,-3,-4,4,9,-3,-12,-3,9,9,4,-3,-12,-3,-12,9,9,-3,12,4,9,
%T A326415 -4,-12,-3,-27,-3,-4,9,9,9,16,-3,9,9,12,-3,-27,-3,-12,-12,9,-3,-12,4,
%U A326415 -12,9,-12,-3,12,9,12,9,9,-3,36,-3,9,-12,4,9,-27,-3,-12,9,-27,-3,-16,-3,9,-12
%N A326415 Dirichlet g.f.: zeta(2*s) / zeta(s)^3.
%C A326415 Moebius transform applied twice to A008836.
%C A326415 Dirichlet inverse of A048691.
%H A326415 Amiram Eldar, <a href="/A326415/b326415.txt">Table of n, a(n) for n = 1..10000</a>
%H A326415 Ilya Gutkovskiy, <a href="/A326415/a326415.jpg">Scatter plot of partial sums of A326415 up to n=10000</a>.
%F A326415 a(n) = Sum_{d|n} mu(n/d) * (-1)^bigomega(d) * 2^omega(d), where mu = A008683, bigomega = A001222 and omega = A001221.
%F A326415 a(1) = 1; a(n) = -Sum_{d|n, d<n} tau((n/d)^2) * a(d), where tau = A000005.
%F A326415 a(n) = Sum_{d|n} A008836(n/d) * A007427(d).
%F A326415 a(n) = Sum_{d|n} A010052(n/d) * A007428(d).
%F A326415 Multiplicative with a(p^e) = -3 if e = 1, and 4*(-1)^e otherwise. - _Amiram Eldar_, Oct 26 2020
%F A326415 b(n) = abs( a(n) ) is multiplicative with b(p) = 3 and b(p^e) = 4 for e > 1 and prime p. Its Dirichlet g.f. is: zeta(s)^3 / zeta(2*s)^2. - _Werner Schulte_, Jan 18 2023
%t A326415 Table[Sum[MoebiusMu[n/d] (-1)^PrimeOmega[d] 2^PrimeNu[d], {d, Divisors[n]}], {n, 1, 75}]
%t A326415 a[n_] := If[n == 1, n, -Sum[If[d < n, DivisorSigma[0, (n/d)^2] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 75}]
%t A326415 f[p_, e_] := If[e == 1, -3, (-1)^e*4]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Oct 26 2020 *)
%Y A326415 Cf. A000005, A001221, A001222, A007427, A007428, A008836, A010052, A034444, A046951, A048691, A158522.
%K A326415 sign,mult,easy
%O A326415 1,2
%A A326415 _Ilya Gutkovskiy_, Oct 18 2019