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Greatest common divisor of largest square dividing n and squarefree part of n.
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%I #40 Oct 08 2022 09:45:03

%S 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,3,1,1,1,1,2,1,1,

%T 1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,2,1,1,1,1,1,1,1,1,1,1,1,1,

%U 1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1

%N Greatest common divisor of largest square dividing n and squarefree part of n.

%C Record values occur at cubes of squarefree numbers: a(A062838(n)) = A005117(n) and a(m) < A005117(n) for m < A062838(n). - _Reinhard Zumkeller_, Apr 09 2010

%H Reinhard Zumkeller, <a href="/A055229/b055229.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = gcd[A008833(n), A007913(n)].

%F Multiplicative with a(p^e)=1 for even e, a(p)=1, a(p^e)=p for odd e>1. - _Vladeta Jovovic_, Apr 30 2002

%F A220218(a(n)) = 1; A060476(a(n)) > 1 for n > 1. - _Reinhard Zumkeller_, Nov 30 2015

%F a(n) = core(n)*rad(n/core(n))/rad(n), where core = A007913 and rad = A007947. - Conjecture by _Velin Yanev_, proof by _David J. Seal_, Sep 19 2017

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} ((p^3 + p^2 + p - 1)/(p^2 * (p + 1))) = 1.2249749939341923764... . - _Amiram Eldar_, Oct 08 2022

%t a[n_] := With[{sf = Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]}& /@ FactorInteger[n])}, GCD[sf, n/sf]]; Table[a[n], {n, 1, 105}] (* _Jean-François Alcover_, Feb 05 2014 *)

%o (PARI) a(n)=my(c=core(n));gcd(c,n/c) \\ _Charles R Greathouse IV_, Nov 20 2012

%o (Haskell)

%o a055229 n = product $ zipWith (^) ps (map (flip mod 2) es) where

%o (ps, es) = unzip $

%o filter ((> 1) . snd) $ zip (a027748_row n) (a124010_row n)

%o -- _Reinhard Zumkeller_, Oct 27 2015

%Y Cf. A000188, A007913, A007947, A008833.

%Y Cf. A027748, A005117, A062838, A060476, A124010, A220218.

%K nice,nonn,mult

%O 1,8

%A _Labos Elemer_, Jun 21 2000