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

1,2

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)

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)

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A278762 A055076 A069780 * A366308 A347089 A354825

Adjacent sequences: A369304 A369305 A369306 * A369308 A369309 A369310

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 19 2024

Status

approved