A370296 Inverse Moebius transform of A322327.

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, 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, 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

Offset

1,2

Links

Table of n, a(n) for n=1..84.

Formula

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

Dirichlet g.f.: (zeta(s))^3 * zeta(2*s) / zeta(4*s).

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.

Dirichlet convolution of A000005 and A323308.

Mathematica

f[p_, e_] := e^2 + e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 14 2024 *)

Prog

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

Crossrefs

Cf. A322327, A000005, A323308.

Sequence in context: A208884 A034257 A145501 * A182139 A092693 A134661

Adjacent sequences: A370293 A370294 A370295 * A370297 A370298 A370299

Keyword

nonn,easy,mult

Author

Werner Schulte, Feb 14 2024

Status

approved