A282672 Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^6.

1, 1138842118714300, 1605078397568386, 1785922862964240, 1878157384495600, 2020105305316098, 2055406015517400, 2071857393746595, 2310442996851990, 2450253379658700, 2513216312053944, 2966830431558840, 2990886595291870, 3228082757486928, 3318987930069240

Offset: 1

Author

Giovanni Resta, Mar 16 2017

Keywords

• nonn

Comments

Also numbers k such that the k-th Catalan number C(2*k,k)/(k+1) is divisible by k^6.

The asymptotic density of this sequence is 3.40390904801... *10^(-13) (Ford and Konyagin, 2021). - Amiram Eldar, Jan 26 2021

Examples

			Let E(n,p) be the exponent of the prime p in the factorization of n. Note that E(n!,p) can be easily found with Legendre's formula without computing n!. Then, t = 1138842118714300 is in the sequence because for each prime p dividing t we have E(C(2*t,t),p) = E((2*t)!,p) - 2*E(t!,p) >= 6*E(t,p).
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..97
• 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, A121943, A282163, A282346, A283073, A283074.