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

33, 35, 51, 57, 65, 75, 85, 95, 105, 115, 117, 119, 129, 135, 147, 171, 175, 183, 185, 201, 219, 221, 225, 235, 237, 245, 247, 253, 255, 261, 279, 285, 291, 295, 301, 309, 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

Offset: 1

Author

Joerg Arndt, Nov 30 2023

Keywords

• nonn

Comments

a(n) is odd since binomial(2*k,k) is even for k>0. - Chai Wah Wu, Nov 30 2023

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Maple

isa := n -> irem(irem(binomial(2*n, n), n), 2) = 1:
select(isa, [seq(1..507, 2)]);  # Peter Luschny, Nov 30 2023

Mathematica

A367782Q[n_]:=OddQ[Mod[Binomial[2n,n],n]];
Select[Range[1000],A367782Q] (* Paolo Xausa, Dec 01 2023 *)

PARI

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

Python

from math import comb
from itertools import count, islice
def A367782_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: (comb(n<<1,n)%n)&1, count(max(startvalue+(startvalue&1^1),1),2))
A367782_list = list(islice(A367782_gen(),30)) # Chai Wah Wu, Nov 30 2023