A389887 - OEIS

%I #25 Dec 10 2025 07:00:31

%S 1,0,0,2,0,0,0,-1,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,-1,0,0,0,0,-2,0,

%T 0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,

%U 0,0,0,0,-2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4

%N Multiplicative sequence a(n) with a(p^e) = (3 + (-1)^e * (2*e+1)) / 4 for prime p and e >= 0.

%C Dirichlet inverse b(n) is multiplicative with b(p^e) = 0 if e = 1, -2 if e mod 3 = 2, and 1 otherwise.

%H Aloe Poliszuk, <a href="/A389887/b389887.txt">Table of n, a(n) for n = 1..10000</a>

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

%F Dirichlet convolution of A008836 and A056624.

%t f[p_, e_] := (3 + (-1)^e*(2*e + 1))/4; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 20 2025 *)

%o (PARI) a(n) = factorback(apply(e -> (3+(-1)^e*(2*e+1))/4,factor(n)[, 2]))

%Y Cf. A008836, A056624.

%K sign,easy,mult

%O 1,4

%A _Werner Schulte_, Nov 20 2025