A369310 The number of divisors d of n such that gcd(d, n/d) is a powerful number.

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, 4, 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, 5, 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, Jan 19 2024

Keywords

• nonn
• easy
• mult

Comments

First differs from A365488 at n = 32, and from A365171 at n = 64.

Links

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

Crossrefs

Cf. A000005, A001620, A001694, A005117, A034444, A046100, A365171, A365488, A369309.

Programs

Mathematica

f[p_, e_] := If[e <= 3, 2, e - 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

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

Python

from math import prod
from sympy import factorint
def A369310(n): return prod(2 if e<=2 else e-1 for e in factorint(n).values()) # Chai Wah Wu, Jan 19 2024

Formula

Multiplicative with a(p^e) = 2 if e <= 3, and e-1 otherwise.

a(n) >= A034444(n), with equality if and only if n is biquadratefree (A046100).

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

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

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

f(1) = Product_{p prime} (1 - 2/p^2 + 1/p^4) = 0.66922021803510257394...,

f'(1)/f(1) = 2 * Sum_{p prime} (p^2-2) * log(p) / (p^4 - p^2 + 1) = 0.81150060034711480230..., and gamma is Euler's constant (A001620).