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

1,2

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

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).

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A365210 A365488 A365171 * A318465 A073180 A316398

Adjacent sequences: A369307 A369308 A369309 * A369311 A369312 A369313

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 19 2024

Status

approved