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

1,2

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

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)).

Mathematica

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

Prog

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

Crossrefs

Cf. A000005, A019555, A053149, A365488.

Sequence in context: A285052 A369719 A369757 * A202686 A134660 A132383

Adjacent sequences: A365486 A365487 A365488 * A365490 A365491 A365492

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 05 2023

Status

approved