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Inverse Moebius transform of A322328.
0

%I #9 Feb 16 2024 09:56:43

%S 1,5,5,13,5,25,5,25,13,25,5,65,5,25,25,41,5,65,5,65,25,25,5,125,13,25,

%T 25,65,5,125,5,61,25,25,25,169,5,25,25,125,5,125,5,65,65,25,5,205,13,

%U 65,25,65,5,125,25,125,25,25,5,325,5,25,65,85,25,125,5,65,25,125,5,325

%N Inverse Moebius transform of A322328.

%F Multiplicative with a(p^e) = 1 + 2*e*(e+1) for prime p and e >= 0.

%F Dirichlet g.f.: (zeta(s))^5 / (zeta(2*s))^2.

%F Dirichlet convolution of A034444 and A048691.

%F Dirichlet inverse sequence b(n) for n > 0 is multiplicative with b(p) = -5 and b(p^e) = (-1)^e * (8*e-4) for prime p and e > 1.

%t f[p_, e_] := 2*e^2 + 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+2*e*(e+1), factor(n)[,2]))

%Y Cf. A034444, A048691, A322328.

%K nonn,easy,mult

%O 1,2

%A _Werner Schulte_, Feb 14 2024