A121943 - OEIS

A121943

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

1, 924, 1287, 2002, 2145, 3366, 3640, 3740, 4199, 6006, 6118, 6552, 7480, 7920, 8580, 8855, 10465, 10920, 11385, 11592, 12285, 12325, 12441, 12540, 12597, 12920, 13224, 13398, 13566, 15080, 15834, 18270, 18354, 18837, 18972, 19227, 23562, 23870, 25641, 25740

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OFFSET

1,2

COMMENTS

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

The asymptotic density of this sequence is 0.00322778... (Ford and Konyagin, 2021). - Amiram Eldar, Jan 26 2021

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Kevin Ford and Sergei Konyagin, Divisibility of the central binomial coefficient binomial(2n, n), Trans. Amer. Math. Soc., Vol. 374, No. 2 (2021), pp. 923-953; arXiv preprint, arXiv:1909.03903 [math.NT], 2019-2020.

MATHEMATICA

Select[Table[n, {n, 20000}], IntegerQ[Binomial[2#, # ]/#^2] &]

PROG

(Python)

from __future__ import division

A121943_list, b = [], 2

for n in range(1, 10**5):

if not b % (n**2):

A121943_list.append(n)

b = b*(4*n+2)//(n+1) # Chai Wah Wu, Mar 27 2016

(PARI) lista(nn) = {for(n=1, nn, if(Mod(binomial(2*n, n), n^2) == 0, print1(n, ", "))); } \\ Altug Alkan, Mar 27 2016

CROSSREFS