A369719 The number of divisors of the smallest cubefull number that is a multiple of n.

1, 4, 4, 4, 4, 16, 4, 4, 4, 16, 4, 16, 4, 16, 16, 5, 4, 16, 4, 16, 16, 16, 4, 16, 4, 16, 4, 16, 4, 64, 4, 6, 16, 16, 16, 16, 4, 16, 16, 16, 4, 64, 4, 16, 16, 16, 4, 20, 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, Jan 30 2024

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A000005, A036966, A356193, A369716, A369720, A369721.

Programs

Mathematica

f[p_, e_] := If[e <= 2, 4, e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecprod(apply(x -> if(x <= 2, 4, x+1), factor(n)[, 2]));

PARI

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X)^2 * ((1 + 2*X - 3*X^2 + X^4)))[n], ", ")) \\ Vaclav Kotesovec, Jan 30 2024

Formula

a(n) = A000005(A356193(n)).

Multiplicative with a(p) = 4 for e <= 2, and a(p^e) = e+1 for e >= 3.

a(n) >= A000005(n), with equality if and only if n is cubefull (A036966).

Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 2/p^s - 3/p^(2*s) + 1/p^(4*s)).

Dirichlet g.f.: zeta(s)^4 * Product_{p prime} (1 + 1/p^(6*s) - 2/p^(5*s) - 2/p^(4*s) + 8/p^(3*s) - 6/p^(2*s)). - Vaclav Kotesovec, Jan 30 2024