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Greatest common divisor of n-th squarefree number and n-th cubefree number.
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%I #19 Oct 20 2023 06:46:36

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

%T 1,2,1,1,1,1,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,1,5,1,1,2,1,1,1,1,1,

%U 2,1,1,1,1,1,1,2,1,1,3,2,1,1,2,1,2,1,3,1,1,1

%N Greatest common divisor of n-th squarefree number and n-th cubefree number.

%C a(n) is squarefree.

%C Conjecture: for any squarefree k exists m such that a(m)=k; see A070078.

%H Amiram Eldar, <a href="/A070077/b070077.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cubefree.html">Cubefree</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree</a>.

%F a(n) = GCD(A005117(n), A004709(n)).

%e a(10) = GCD(A005117(10), A004709(10)) = GCD(14,11) = 1.

%e a(11) = GCD(A005117(11), A004709(11)) = GCD(15,12) = 3*GCD(5,4) = 3*1 = 3.

%t With[{max = 150}, e = Max[FactorInteger[#][[;; , 2]]] & /@ Range[max]; sqf = Flatten[Position[e, _?(# < 2 &)]]; cbf = Flatten[Position[e, _?(# < 3 &)]]; GCD[sqf, cbf[[1 ;; Length[sqf]]]]] (* _Amiram Eldar_, Oct 20 2023 *)

%Y Cf. A004709, A005117, A070078.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Apr 22 2002