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

1, 2, 4, 6, 8, 20, 32, 54, 66, 110, 144, 170, 200, 210, 278, 288, 304, 330, 402, 405, 468, 510, 527, 628, 654, 684, 704, 778, 783, 784, 853, 891, 892, 990, 1001, 1035, 1125, 1155, 1232, 1296, 1334, 1384, 1394, 1488, 1495, 1521, 1551, 1575, 1600, 1625, 1645, 1701, 1768, 1875, 1891, 2028, 2072

Offset: 1

Author

Robert Israel, May 26 2020

Keywords

• nonn

Comments

Numbers k such that A020733(2*k) = 1.

Examples

			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.
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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..250

Crossrefs

Cf. A001221, A020733, A067434.

Programs

Maple

filter:=  proc(n) local t, v, i, m;
  m:= 0: t:= 1:
  for i from 1 to n-1 do
   t:= t * ifactor(2*n-i+1)/ifactor(i);
   if type(t,`*`) then v:= nops(t) else v:= 1 fi;
   if v > m then m:= v fi;
  od;
  t:= t*ifactor(n+1)/ifactor(n);
  type(t,`*`) and nops(t) > m
end proc:
filter(1):= true:
select(filter, [$1..2500]); # Robert Israel, May 26 2020

Mathematica

Select[Range@ 1001, Max@ Most@ # < Last@ # &@ PrimeNu@ Binomial[2 #, Range[0, #]] &] (* Michael De Vlieger, May 26 2020 *)