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

Offset: 1

Author

Tanya Khovanova, Sep 03 2006

Keywords

• nonn

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.

Crossrefs

Cf. A000108, A000984, A014847, A282163, A282346, A283073, A283074, A282672.

Programs

Mathematica

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

PARI

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

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