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A335209 Numbers k such that binomial(2*k,k) has more distinct prime factors than binomial(2*k,i) for 0 <= i < k.

%I A335209 #18 Jun 26 2021 18:03:59
%S A335209 1,2,4,6,8,20,32,54,66,110,144,170,200,210,278,288,304,330,402,405,
%T A335209 468,510,527,628,654,684,704,778,783,784,853,891,892,990,1001,1035,
%U A335209 1125,1155,1232,1296,1334,1384,1394,1488,1495,1521,1551,1575,1600,1625,1645,1701,1768,1875,1891,2028,2072
%N A335209 Numbers k such that binomial(2*k,k) has more distinct prime factors than binomial(2*k,i) for 0 <= i < k.
%C A335209 Numbers k such that A020733(2*k) = 1.
%H A335209 Robert Israel, <a href="/A335209/b335209.txt">Table of n, a(n) for n = 1..250</a>
%e A335209 a(4)=6 is in the sequence because binomial(12,6) = 924 = 2^2*3*7*11 has 4 distinct prime factors while binomial(12,0) to binomial(12,5) all have at most 3.
%e A335209 7 is not in the sequence because binomial(14,7) = 3432 = 2^3*3*11*13 and binomial(14,6) = 3003 = 3*7*11*13 both have 4 distinct prime factors.
%p A335209 filter:=  proc(n) local t, v, i, m;
%p A335209   m:= 0: t:= 1:
%p A335209   for i from 1 to n-1 do
%p A335209    t:= t * ifactor(2*n-i+1)/ifactor(i);
%p A335209    if type(t,`*`) then v:= nops(t) else v:= 1 fi;
%p A335209    if v > m then m:= v fi;
%p A335209   od;
%p A335209   t:= t*ifactor(n+1)/ifactor(n);
%p A335209   type(t,`*`) and nops(t) > m
%p A335209 end proc:
%p A335209 filter(1):= true:
%p A335209 select(filter, [$1..2500]); # _Robert Israel_, May 26 2020
%t A335209 Select[Range@ 1001, Max@ Most@ # < Last@ # &@ PrimeNu@ Binomial[2 #, Range[0, #]] &] (* _Michael De Vlieger_, May 26 2020 *)
%Y A335209 Cf. A001221, A020733, A067434.
%K A335209 nonn
%O A335209 1,2
%A A335209 _Robert Israel_, May 26 2020