A389887 Multiplicative sequence a(n) with a(p^e) = (3 + (-1)^e * (2*e+1)) / 4 for prime p and e >= 0.

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, 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, 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

Offset

1,4

Comments

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

Links

Aloe Poliszuk, Table of n, a(n) for n = 1..10000

Formula

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

Dirichlet convolution of A008836 and A056624.

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A008836, A056624.

Sequence in context: A132406 A197881 A332712 * A388711 A079126 A339086

Adjacent sequences: A389884 A389885 A389886 * A389888 A389889 A389890

Keyword

sign,easy,mult

Author

Werner Schulte, Nov 20 2025

Status

approved