A369719 - OEIS

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

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OFFSET

1,2

LINKS

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

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

MATHEMATICA

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

PROG

(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

CROSSREFS

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

Sequence in context: A107680 A358509 A285052 * A369757 A365489 A202686

Adjacent sequences: A369716 A369717 A369718 * A369720 A369721 A369722

KEYWORD

nonn,easy,mult

AUTHOR

Amiram Eldar, Jan 30 2024

STATUS

approved