A380922 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s + 1/p^(3*s)).

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

Offset: 1

Author

Vaclav Kotesovec, Apr 22 2025

Keywords

• nonn
• mult
• easy

Comments

First differs from A061389 at n = 32.

First differs from A322483 at n = 32.

First differs from A372380 at n = 128 (next differences are at n=128*k, n=2187*k, ...).

The number of divisors of n that are both biquadratefree (A046100) and exponentially odd (A268335), i.e., in A336591. - Amiram Eldar, Apr 22 2025

Links

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

Crossrefs

Cf. A073184, A365498, A383292.

Cf. A322483, A372380, A061389.

Cf. A046100, A268335, A336591.

Programs

Mathematica

f[p_, e_] := If[e < 3, 2, 3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 22 2025 *)

PARI

for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * (1 + X + X^3))[n], ", "))

Formula

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

Dirichlet g.f.: zeta(s)^2 * f(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 - 1/p^2 + 1/p^3 - 1/p^4) = 0.684286924186862318141968725791218083472312736723163777284618226290055...,

f'(1) = f(1) * Sum_{p prime} (2*p^2 - 3*p + 4) * log(p) / ((p-1) * (p^3 + p^2 + 1)) = f(1) * 0.85825768698295295413525347933038488513032293516964600096226328323449...

and gamma is the Euler-Mascheroni constant A001620.

Multiplicative with a(p^e) = 2 if e <= 2 and 3 otherwise. - Amiram Eldar, Apr 22 2025