A369307 The number of exponentially odd divisors d of n such that n/d is also exponentially odd.

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

Offset: 1

Author

Amiram Eldar, Jan 19 2024

Keywords

• nonn
• easy
• mult

Comments

First differs from A366308 at n = 32.

Dirichlet convolution of A295316 with itself.

Links

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

Crossrefs

Cf. A000005, A005117, A062503, A268335, A295316, A322483, A365173, A366308, A369308.

Programs

Mathematica

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

PARI

a(n) = vecprod(apply(x -> if(x%2, 2, x/2), factor(n)[,2]));

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 - X^2 + X)^2/(1 - X^2)^2)[n], ", ")) \\ Vaclav Kotesovec, Jan 19 2024

Python

from math import prod
from sympy import factorint
def A369307(n): return prod(2 if e&1 else e>>1 for e in factorint(n).values()) # Chai Wah Wu, Jan 19 2024

Formula

Multiplicative with a(p^e) = 2 is e is odd, and e/2 if e is even.

a(n) >= 1, with equality if and only if n is the square of a squarefree number (A062503).

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

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

From Vaclav Kotesovec, Jan 19 2024: (Start)

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

Let f(s) = Product_{p prime} (p^(2*s) + p^s - 1)^2 / ((p^s + 1)^2 * p^(2*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 + 2*p - 1) / (p^2*(p+1)^2)) = 0.49623881454854881762168565097162197963340069996226074849602334089041678...,

f'(1) = f(1) * Sum_{p prime} 2*(2*p + 1) * log(p) / ((p+1)*(p^2 + p - 1)) = f(1) * 1.49674466685934940187617305887881799198585080518913793200171026177150513...

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