%I #13 Jul 21 2022 05:31:21

%S 1,2,3,4,5,6,8,9,10,12,16,18,20,24,32,33,34,36,40,48,65,66,72,96,136,

%T 144,192,256,258,264,288,520,576,768,1056

%N Numbers n such that binomial(2n,n) is cubefree.

%C No others < 15000. This sequence is probably complete. According to Sander, this sequence - and the sequence for any power - is finite.

%C No more terms < 10^8. - _Bert Dobbelaere_, Jul 21 2022

%H J. W. Sander, <a href="http://www.jstor.org/stable/2974508">A story of binomial coefficients and primes</a>, Amer. Math. Monthly 102 (1995), 802-807.

%t t3=Table[f=FactorInteger[Binomial[2n, n]]; s=Select[f, #[[2]]>2&]; If[s=={}, 0, s[[ -1, 1]]], {n, 15000}]; Flatten[Position[t3, 0]]

%o (PARI) isok(n) = vecmax(factor(binomial(2*n, n))[,2]) < 3; \\ _Michel Marcus_, Oct 04 2017

%Y Cf. A110496 (least k such that prime(n)^3 divides binomial(2k, k)).

%K nonn,fini,more

%O 1,2

%A _T. D. Noe_, Jul 22 2005