A367782 - OEIS

%I #22 Dec 01 2023 04:36:05

%S 33,35,51,57,65,75,85,95,105,115,117,119,129,135,147,171,175,183,185,

%T 201,219,221,225,235,237,245,247,253,255,261,279,285,291,295,301,309,

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

%C a(n) is odd since binomial(2*k,k) is even for k>0. - _Chai Wah Wu_, Nov 30 2023

%H Paolo Xausa, <a href="/A367782/b367782.txt">Table of n, a(n) for n = 1..10000</a>

%p isa := n -> irem(irem(binomial(2*n, n), n), 2) = 1:

%p select(isa, [seq(1..507, 2)]);  # _Peter Luschny_, Nov 30 2023

%t A367782Q[n_]:=OddQ[Mod[Binomial[2n,n],n]];

%t Select[Range[1000],A367782Q] (* _Paolo Xausa_, Dec 01 2023 *)

%o (PARI) for(n=1,510,if(bitand(binomial(2*n,n)%n,1),print1(n,", ")));

%o (Python)

%o from math import comb

%o from itertools import count, islice

%o def A367782_gen(startvalue=1): # generator of terms >= startvalue

%o     return filter(lambda n: (comb(n<<1,n)%n)&1, count(max(startvalue+(startvalue&1^1),1),2))

%o A367782_list = list(islice(A367782_gen(),30)) # _Chai Wah Wu_, Nov 30 2023

%Y Cf. A059288 (binomial(2*n,n) mod n), A014847 (k such that binomial(2*k,k) mod k is zero).

%K nonn

%O 1,1

%A _Joerg Arndt_, Nov 30 2023