A389831 Multiplicative sequence a(n) with a(p^e) = 2 + e^2 for prime p and e > 0.

1, 3, 3, 6, 3, 9, 3, 11, 6, 9, 3, 18, 3, 9, 9, 18, 3, 18, 3, 18, 9, 9, 3, 33, 6, 9, 11, 18, 3, 27, 3, 27, 9, 9, 9, 36, 3, 9, 9, 33, 3, 27, 3, 18, 18, 9, 3, 54, 6, 18, 9, 18, 3, 33, 9, 33, 9, 9, 3, 54, 3, 9, 18, 38, 9, 27, 3, 18, 9, 27, 3, 66, 3, 9, 18, 18, 9, 27, 3, 54, 18, 9, 3, 54

Offset

1,2

Comments

Dirichlet inverse b(n) is multiplicative with b(p^e) = (2 + e^2 mod 3) * (-1)^e for prime p and e > 0. - Werner Schulte, Oct 25 2025

Links

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Formula

Dirichlet g.f.: (zeta(s))^3 * zeta(3*s) / zeta(6*s).

Dirichlet convolution of A005361 and A034444.

Sum_{k=1..n} a(k) ~ (945*n/(2*Pi^6)) * (log(n)^2 * zeta(3) + 2*log(n)*((3*gamma - 1)*zeta(3) + 3*zeta'(3) - 5670*zeta(3)*(zeta'(6)/Pi^6)) + (2 + 6*(gamma - 1)*gamma - 6*sg1)*zeta(3) + 6*(3*gamma - 1)*zeta'(3) + 9*zeta''(3) - 11340*((((3*gamma - 1)*zeta(3) + 3*zeta'(3))* zeta'(6) + 3*zeta(3)*zeta''(6))/Pi^6) + 64297800*zeta(3)*zeta'(6)^2/Pi^12), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Oct 19 2025

Maple

a:= n-> mul(2+i[2]^2, i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 16 2025

Mathematica

f[p_, e_] := e^2 + 2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 16 2025 *)

Prog

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

Crossrefs

Cf. A005361, A034444.

Sequence in context: A307000 A007425 A260152 * A358223 A130695 A308083

Adjacent sequences: A389828 A389829 A389830 * A389832 A389833 A389834

Keyword

nonn,easy,mult

Author

Werner Schulte, Oct 16 2025

Status

approved