%I #13 Feb 15 2024 13:02:13
%S 1,3,3,7,3,9,3,13,7,9,3,21,3,9,9,21,3,21,3,21,9,9,3,39,7,9,13,21,3,27,
%T 3,31,9,9,9,49,3,9,9,39,3,27,3,21,21,9,3,63,7,21,9,21,3,39,9,39,9,9,3,
%U 63,3,9,21,43,9,27,3,21,9,27,3,91,3,9,21,21,9,27,3,63,21,9,3,63
%N Inverse Moebius transform of A322327.
%F Multiplicative with a(p^e) = 1 + e + e^2 for prime p and e >= 0.
%F Dirichlet g.f.: (zeta(s))^3 * zeta(2*s) / zeta(4*s).
%F Dirichlet inverse sequence b(n) for n > 0 is multiplicative with b(p) = -3 and b(p^e) = 2 * (-1)^((e+1)*(e+2)/2) for prime p and e > 1.
%F Dirichlet convolution of A000005 and A323308.
%t f[p_, e_] := e^2 + e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Feb 14 2024 *)
%o (PARI) a(n) = factorback(apply(e->1+e+e^2,factor(n)[,2]))
%Y Cf. A322327, A000005, A323308.
%K nonn,easy,mult
%O 1,2
%A _Werner Schulte_, Feb 14 2024