A071705 - OEIS

%I #20 Apr 02 2022 16:39:54

%S 1,2,5,9,13,20,18,21,20,63,50,131,111,67,197,113,113,338,335,173,426,

%T 110,110,554,515,515,368,368,515,928,928,1269,1152,1152,1269,1511,

%U 1462,1456,1458,1458,2524,2181,2895,2895,2895,2805,3379,3379,3640,2808,3284

%N Least k > n such that C(2n,n) divides C(2k,k).

%C Erdős proved that a(n) >= 2n, and that there is a constant c > 0 such that for sufficient large n, n^(1+c) < a(n) < (2n)^(log(n)/log(2)). - _Amiram Eldar_, May 18 2017

%H Giovanni Resta, <a href="/A071705/b071705.txt">Table of n, a(n) for n = 0..600</a>

%H Paul Erdős, <a href="http://dx.doi.org/10.4153/CMB-1964-047-5">On some divisibility properties of (2n n)</a>, Canadian Mathematical Bulletin, Vol. 7, No. 4 (1964), pp. 513-518.

%t f[n_] := Block[{k = n + 1}, bn = Binomial[2n, n]; While[ !IntegerQ[ Binomial[2k, k]/bn], k++ ]; k]; Table[ f[n], {n, 0, 50}]

%t lk[n_]:=Module[{k=n+1,c=Binomial[2n,n]},While[Mod[Binomial[2k,k],c]!=0,k++];k]; Array[lk,60,0] (* _Harvey P. Dale_, Apr 02 2022 *)

%o (PARI) for(n=1,45,s=n+1; while(binomial(2*s,s)%binomial(2*n,n)>0,s++); print1(s,","))

%K nonn

%O 0,2

%A _Benoit Cloitre_, Jun 24 2002

%E Edited by _Robert G. Wilson v_, Jun 27 2002