A383292 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(2*s) + 1/p^(3*s)).

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 3, 1, 2, 2, 4

Offset: 1

Author

Vaclav Kotesovec, Apr 22 2025

Keywords

• nonn
• mult
• easy

Comments

First differs from A095691, A365552 and A368105 at n = 32.

The number of divisors of n that are both biquadratefree (A046100) and powerful (A001694). - Amiram Eldar, Apr 22 2025

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A073184, A365498, A380922.

Cf. A095691, A365552, A368105.

Cf. A330595.

Cf. A046100, A001694.

Programs

Mathematica

f[p_, e_] := If[e < 4, e, 3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 22 2025 *)

PARI

for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * (1 + X^2 + X^3))[n], ", "))

Formula

Sum_{k=1..n} a(k) ~ c * n, where c = A330595 = Product_{p prime} (1 + 1/p^2 + 1/p^3) = 1.74893299784324530303390699768511480225988349359548...

Multiplicative with a(p^e) = e if e < 4 and 3 otherwise. - Amiram Eldar, Apr 22 2025