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A367782 Numbers k such that binomial(2*k,k) mod k is odd.

%I A367782 #22 Dec 01 2023 04:36:05
%S A367782 33,35,51,57,65,75,85,95,105,115,117,119,129,135,147,171,175,183,185,
%T A367782 201,219,221,225,235,237,245,247,253,255,261,279,285,291,295,301,309,
%U A367782 319,329,333,335,341,357,365,369,377,381,385,395,399,403,415,417,423,427,453,455,471,473,481,485,489,507
%N A367782 Numbers k such that binomial(2*k,k) mod k is odd.
%C A367782 a(n) is odd since binomial(2*k,k) is even for k>0. - _Chai Wah Wu_, Nov 30 2023
%H A367782 Paolo Xausa, <a href="/A367782/b367782.txt">Table of n, a(n) for n = 1..10000</a>
%p A367782 isa := n -> irem(irem(binomial(2*n, n), n), 2) = 1:
%p A367782 select(isa, [seq(1..507, 2)]);  # _Peter Luschny_, Nov 30 2023
%t A367782 A367782Q[n_]:=OddQ[Mod[Binomial[2n,n],n]];
%t A367782 Select[Range[1000],A367782Q] (* _Paolo Xausa_, Dec 01 2023 *)
%o A367782 (PARI) for(n=1,510,if(bitand(binomial(2*n,n)%n,1),print1(n,", ")));
%o A367782 (Python)
%o A367782 from math import comb
%o A367782 from itertools import count, islice
%o A367782 def A367782_gen(startvalue=1): # generator of terms >= startvalue
%o A367782     return filter(lambda n: (comb(n<<1,n)%n)&1, count(max(startvalue+(startvalue&1^1),1),2))
%o A367782 A367782_list = list(islice(A367782_gen(),30)) # _Chai Wah Wu_, Nov 30 2023
%Y A367782 Cf. A059288 (binomial(2*n,n) mod n), A014847 (k such that binomial(2*k,k) mod k is zero).
%K A367782 nonn
%O A367782 1,1
%A A367782 _Joerg Arndt_, Nov 30 2023