A365489 The number of divisors of the smallest cube divisible by n.

1, 4, 4, 4, 4, 16, 4, 4, 4, 16, 4, 16, 4, 16, 16, 7, 4, 16, 4, 16, 16, 16, 4, 16, 4, 16, 4, 16, 4, 64, 4, 7, 16, 16, 16, 16, 4, 16, 16, 16, 4, 64, 4, 16, 16, 16, 4, 28, 4, 16, 16, 16, 4, 16, 16, 16, 16, 16, 4, 64, 4, 16, 16, 7, 16, 64, 4, 16, 16, 64, 4, 16, 4

Offset: 1

Author

Amiram Eldar, Sep 05 2023

Keywords

• nonn
• easy
• mult

Comments

The number of divisors of the cube root of the smallest cube divisible by n, A019555(n), is A365488(n).

Links

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

Crossrefs

Cf. A000005, A019555, A053149, A365345, A365488.

Programs

Mathematica

f[p_, e_] := 3*Ceiling[e/3] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecprod(apply(x -> 3*((x-1)\3) + 4, factor(n)[, 2]));

Formula

a(n) = A000005(A053149(n)).

Multiplicative with a(p^e) = 3*ceiling(e/3) + 1.

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