A282672 - OEIS

%I #34 Jan 26 2021 04:02:07

%S 1,1138842118714300,1605078397568386,1785922862964240,

%T 1878157384495600,2020105305316098,2055406015517400,2071857393746595,

%U 2310442996851990,2450253379658700,2513216312053944,2966830431558840,2990886595291870,3228082757486928,3318987930069240

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

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

%C The asymptotic density of this sequence is 3.40390904801... *10^(-13) (Ford and Konyagin, 2021). - _Amiram Eldar_, Jan 26 2021

%H Giovanni Resta, <a href="/A282672/b282672.txt">Table of n, a(n) for n = 1..97</a>

%H 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.

%e 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).

%Y Cf. A000108, A000984, A014847, A121943, A282163, A282346, A283073, A283074.

%K nonn

%O 1,2

%A _Giovanni Resta_, Mar 16 2017