cp's OEIS Frontend

A121943 Numbers k such that the central binomial coefficient C(2k,k) is divisible by k^2.

%I A121943 #54 Jan 10 2022 05:28:27
%S A121943 1,924,1287,2002,2145,3366,3640,3740,4199,6006,6118,6552,7480,7920,
%T A121943 8580,8855,10465,10920,11385,11592,12285,12325,12441,12540,12597,
%U A121943 12920,13224,13398,13566,15080,15834,18270,18354,18837,18972,19227,23562,23870,25641,25740
%N A121943 Numbers k such that the central binomial coefficient C(2k,k) is divisible by k^2.
%C A121943 Equivalently, numbers n such that the n-th Catalan number C(2n,n)/(n+1) is divisible by n^2. - _Lucian Craciun_, Feb 09 2017
%C A121943 The asymptotic density of this sequence is 0.00322778... (Ford and Konyagin, 2021). - _Amiram Eldar_, Jan 26 2021
%H A121943 Chai Wah Wu, <a href="/A121943/b121943.txt">Table of n, a(n) for n = 1..10000</a>
%H A121943 Kevin Ford and Sergei Konyagin, <a href="https://doi.org/10.1090/tran/8183">Divisibility of the central binomial coefficient binomial(2n, n)</a>, Trans. Amer. Math. Soc., Vol. 374, No. 2 (2021), pp. 923-953; <a href="https://arxiv.org/abs/1909.03903">arXiv preprint</a>, arXiv:1909.03903 [math.NT], 2019-2020.
%t A121943 Select[Table[n, {n, 20000}], IntegerQ[Binomial[2#, # ]/#^2] &]
%o A121943 (Python)
%o A121943 from __future__ import division
%o A121943 A121943_list, b = [], 2
%o A121943 for n in range(1,10**5):
%o A121943     if not b % (n**2):
%o A121943         A121943_list.append(n)
%o A121943     b = b*(4*n+2)//(n+1) # _Chai Wah Wu_, Mar 27 2016
%o A121943 (PARI) lista(nn) = {for(n=1, nn, if(Mod(binomial(2*n, n), n^2) == 0, print1(n, ", ")));} \\ _Altug Alkan_, Mar 27 2016
%Y A121943 Cf. A000108, A000984, A014847, A282163, A282346, A283073, A283074, A282672.
%K A121943 nonn
%O A121943 1,2
%A A121943 _Tanya Khovanova_, Sep 03 2006