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A336643 Squarefree kernel of n divided by the squarefree part of n: a(n) = rad(n) / core(n). 10

%I #36 Sep 09 2023 07:17:09

%S 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,

%T 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,

%U 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

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

%C 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

%H Antti Karttunen, <a href="/A336643/b336643.txt">Table of n, a(n) for n = 1..65537</a>

%H Vaclav Kotesovec, <a href="/A336643/a336643.jpg">Graph - the asymptotic ratio (1000000 terms)</a>.

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

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

%F a(n) = n/A350390(n). - _Amiram Eldar_, Jan 01 2022

%F a(n) = A356191(n)/n. - _Amiram Eldar_, Nov 18 2022

%F 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

%F From _Vaclav Kotesovec_, Sep 09 2023: (Start)

%F 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)).

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

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

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

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

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

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

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

%o (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

%o (Python)

%o from math import prod

%o from sympy.ntheory.factor_ import primefactors, core

%o def A336643(n): return prod(primefactors(n))//core(n) # _Chai Wah Wu_, Dec 30 2021

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

%K nonn,easy,mult

%O 1,4

%A _Antti Karttunen_, Jul 28 2020

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Last modified July 21 02:16 EDT 2024. Contains 374462 sequences. (Running on oeis4.)