A283073 Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^4.

1, 227736432, 338956200, 386160984, 482213160, 544508118, 548823405, 715592220, 726922482, 731987190, 1427877360, 1448431600, 1467104760, 1490842353, 1491241258, 1504640335, 1646570115, 1852712100, 1923506200, 1923927460, 1924947570, 2056580995, 2064409413

Offset: 1

Author

Lucian Craciun, Feb 28 2017

Keywords

• nonn

Comments

Equivalently, numbers k such that the k-th Catalan number C(2*k,k)/(k+1) is divisible by k^4.

The asymptotic density of this sequence is 1.330129946... * 10^(-7) (Ford and Konyagin, 2021). - Amiram Eldar, Jan 26 2021

Examples

			The central binomial coefficient C(2*227736432,227736432) is divisible by 227736432^4.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..1002
• 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.
• Wikipedia Mathematics Reference Desk, n^4 Divides Central Binomial Coefficient.

Crossrefs

Cf. A000108, A000984, A014847, A121943, A282163, A282346, A283074, A282672.

Programs

Mathematica

A283073:={}; k:=4; For[n:=1, n<=10^9, n++, {f=FactorInteger[n], For[j:=1, j<=Length[f], j++, {b=True, If[Sum[Floor[2n/f[[j, 1]]^i]-2 Floor[n/f[[j, 1]]^i], {i, 1, Length[IntegerDigits[2n, f[[j, 1]]]]}]A283073=Append[A283073, n]]}] (* Legendre's formula for drastic time reduction *)

Extensions

a(11)-a(22) from Giovanni Resta, Feb 28 2017