A365171 The number of divisors d of n such that gcd(d, n/d) is a square.

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, 4, 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, Aug 25 2023

Keywords

• nonn
• easy
• mult

Comments

The sum of these divisors is A365172(n).

Links

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

Crossrefs

Cf. A000005, A000290, A000583, A004524, A005117, A008833, A034444, A046101, A046951, A182448, A365172, A365173.

Programs

Mathematica

f[p_, e_] := Floor[(e + 3)/4] + Floor[(e + 4)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecprod(apply(x -> (x+3)\4 + (x+4)\4, factor(n)[, 2]));

PARI

for(n=1, 100, print1(direuler(p=2, n, 1/((1 - X)^2 * (1 + X^2)))[n], ", ")) \\ Vaclav Kotesovec, Jan 20 2024

Formula

Multiplicative with a(p^e) = floor((e + 3)/4) + floor((e + 4)/4) = A004524(e+3).

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

a(n) >= A034444(n), with equality if and only if n is not a biquadrateful number (A046101).

a(n) == 1 (mod 2) if and only if n is a fourth power (A000583).

From Vaclav Kotesovec, Jan 20 2024: (Start)

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

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

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 - 1/(p^2 + 1)) = Pi^2/15 = A182448,

f'(1) = f(1) * Sum_{p prime} 2*log(p) / (p^2 + 1) = f(1) * 0.8852429263675811068149340172820329246145172848406469350087715037483367369...

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