A368247 The number of cubefree divisors of the cubefull part of n (A360540).

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Dec 19 2023

Keywords

• nonn
• easy
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A004709, A036966, A073184, A323308, A360540.

Programs

Mathematica

f[p_, e_] := If[e > 2, 3, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecprod(apply(x -> if(x < 3, 1, 3), factor(n)[, 2]));

Formula

a(n) = A073184(A360540(n)).

Multiplicative with a(p^e) = 1 if e <= 2, and 3 otherwise.

a(n) >= 1, with equality if and only if n is cubefree (A004709).

a(n) <= A073184(n), with equality if and only if n is cubefull (A036966).

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

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 2/p^3) = 1.37700168952903630206... .

In general, the asymptotic mean of the number of k-free divisors of the k-full part of n is Product_{p prime} (1 + (k-1)/p^k).