A365488 The number of divisors of the smallest number whose cube is divisible by n.

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 3, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 6, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 3, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 4, 8, 2, 6, 3, 4, 2, 8, 4, 4, 4

Offset: 1

Author

Amiram Eldar, Sep 05 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A365171 at n = 32.

The number of divisors of the smallest cube divisible by n, A053149(n), is A365489(n).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Crossrefs

Cf. A000005, A005117, A019555, A053149, A365171, A365489, A365498.

Programs

Mathematica

f[p_, e_] := Ceiling[e/3] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
With[{c=Range[200]^3},Table[DivisorSigma[0,Surd[SelectFirst[c,Mod[#,n]==0&],3]],{n,90}]] (* Harvey P. Dale, Sep 15 2024 *)

PARI

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

Formula

a(n) = A000005(A019555(n)).

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

a(n) <= A000005(n) with equality if and only if n is squarefree (A005117).

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

From Vaclav Kotesovec, Sep 06 2023: (Start)

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

Let f(s) = Product_{p prime} (1 - 1/p^(2*s) - 1/p^(3*s) + 1/p^(4*s)).

Sum_{k=1..n} a(k) ~ zeta(3) * f(1) * n * (log(n) + 2*gamma - 1 + 3*zeta'(3)/zeta(3) + f'(1)/f(1)), where

f(1) = Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085026313185459506482223745141452711510108346133288...,

f'(1) = f(1) * Sum_{p prime} (-4 + 3*p + 2*p^2) * log(p) / (1 - p - p^2 + p^4) = f(1) * 1.4525924794451595590371439593828547341482465114411929136723476679...

and gamma is the Euler-Mascheroni constant A001620. (End)