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Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^4.
7

%I #43 Jan 10 2022 05:26:52

%S 1,227736432,338956200,386160984,482213160,544508118,548823405,

%T 715592220,726922482,731987190,1427877360,1448431600,1467104760,

%U 1490842353,1491241258,1504640335,1646570115,1852712100,1923506200,1923927460,1924947570,2056580995,2064409413

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

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

%C The asymptotic density of this sequence is 1.330129946... * 10^(-7) (Ford and Konyagin, 2021). - _Amiram Eldar_, Jan 26 2021

%H Giovanni Resta, <a href="/A283073/b283073.txt">Table of n, a(n) for n = 1..1002</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.

%H Wikipedia Mathematics Reference Desk, <a href="http://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2017_February_9#n4_divides_Central_binomial_coefficient">n^4 Divides Central Binomial Coefficient</a>.

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

%t 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]]]]}]<f[[j, 2]]k, {b=False, Break[]}]}], If[b, A283073=Append[A283073, n]]}] (* Legendre's formula for drastic time reduction *)

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

%K nonn

%O 1,2

%A _Lucian Craciun_, Feb 28 2017

%E a(11)-a(22) from _Giovanni Resta_, Feb 28 2017