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A055229 Greatest common divisor of largest square dividing n and squarefree part of n.

%I A055229 #40 Oct 08 2022 09:45:03
%S A055229 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 A055229 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 A055229 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 A055229 Greatest common divisor of largest square dividing n and squarefree part of n.
%C A055229 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 A055229 Reinhard Zumkeller, <a href="/A055229/b055229.txt">Table of n, a(n) for n = 1..10000</a>
%F A055229 a(n) = gcd[A008833(n), A007913(n)].
%F A055229 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 A055229 A220218(a(n)) = 1; A060476(a(n)) > 1 for n > 1. - _Reinhard Zumkeller_, Nov 30 2015
%F A055229 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 A055229 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 A055229 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 A055229 (PARI) a(n)=my(c=core(n));gcd(c,n/c) \\ _Charles R Greathouse IV_, Nov 20 2012
%o A055229 (Haskell)
%o A055229 a055229 n = product $ zipWith (^) ps (map (flip mod 2) es) where
%o A055229    (ps, es) = unzip $
%o A055229               filter ((> 1) . snd) $ zip (a027748_row n) (a124010_row n)
%o A055229 -- _Reinhard Zumkeller_, Oct 27 2015
%Y A055229 Cf. A000188, A007913, A007947, A008833.
%Y A055229 Cf. A027748, A005117, A062838, A060476, A124010, A220218.
%K A055229 nice,nonn,mult
%O A055229 1,8
%A A055229 _Labos Elemer_, Jun 21 2000