A336643 Squarefree kernel of n divided by the squarefree part of n: a(n) = rad(n) / core(n).

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 7, 5, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 7, 3, 10, 1, 1, 1, 1, 1

Offset: 1

Author

Antti Karttunen, Jul 28 2020

Keywords

• nonn
• easy
• mult
• changed

Comments

a(n) is the least number k such that k*n (and also n/k) is an exponentially odd number (A268335). - Amiram Eldar, Nov 18 2022

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms).

Crossrefs

Cf. A007913, A007947, A059841, A268335, A336644, A350390, A356191.

Programs

Mathematica

f[p_, e_] := p^(1 - Mod[e, 2]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)

PARI

A336643(n) = (factorback(factorint(n)[, 1]) / core(n));

PARI

A336643(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^(1-(f[i, 2]%2))));

PARI

for(n=1, 100, print1(direuler(p=2, n, 1/(1-X^2) * (1 + X + p*X^2 - X^2))[n], ", ")) \\ Vaclav Kotesovec, Sep 09 2023

Python

from math import prod
from sympy.ntheory.factor_ import primefactors, core
def A336643(n): return prod(primefactors(n))//core(n) # Chai Wah Wu, Dec 30 2021

SageMath

def A336643(n: int) -> int:
    return prod(b^(1 - e % 2) for (b, e) in list(factor(n)))
print([A336643(n) for n in range(1, 106)])  # Peter Luschny, Aug 23 2025

Formula

a(n) = A007947(n) / A007913(n).

Multiplicative with a(p^k) = p^(1-(k mod 2)) = p^A059841(k).

a(n) = n/A350390(n). - Amiram Eldar, Jan 01 2022

a(n) = A356191(n)/n. - Amiram Eldar, Nov 18 2022

Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s-1) - 1/p^(2*s)). - Amiram Eldar, Sep 09 2023

From Vaclav Kotesovec, Sep 09 2023: (Start)

Let f(s) = Product_{p prime} (1 - p^(1-5*s) + p^(2-5*s) + 2*p^(1-4*s) - p^(2-4*s) - p^(1-3*s) + p^(-3*s) - 2*p^(-2*s)).

Dirichlet g.f.: zeta(s) * zeta(2*s) * zeta(2*s-1) * f(s).

Sum_{k=1..n} a(k) ~ Pi^2 * f(1) * n / 12 * (log(n) + 3*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where

f(1) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = A256392 = 0.217778716619536378323007514119446813130797755001355937648276403523626491...,

f'(1) = f(1) * Sum_{p prime} (11*p - 5) * log(p) / (p^3 + p^2 - 3*p + 1) = f(1) * 4.716596820856763078660955244870812634072512131626849517007098664560806248...

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