A042943 Numbers k such that binomial(2^k, k) is divisible by binomial(2^k, 2).

1, 2, 3, 5, 7, 9, 11, 13, 14, 17, 19, 22, 23, 25, 26, 27, 29, 31, 33, 35, 37, 38, 39, 41, 43, 45, 46, 47, 49, 50, 51, 53, 55, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 79, 81, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 99, 101, 102, 103, 106, 107, 109, 111

Offset: 1

Author

Labos Elemer, Apr 11 2001

Keywords

• easy
• nonn

Comments

Does not contain multiples of 4 (A008586).

Crossrefs

Cf. A014070, A006516.

Programs

Mathematica

Select[Range[150],Divisible[Binomial[2^#,#],Binomial[2^#,2]]&]  (* Harvey P. Dale, Mar 24 2011 *)

PARI

isok(k) = (binomial(2^k, k) % binomial(2^k, 2)) == 0; \\ Michel Marcus, May 14 2018

Python

from math import comb
from itertools import count, islice
def A042943_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue,1)):
        if comb(m:=1<A042943_list = list(islice(A042943_gen(),30)) # Chai Wah Wu, Jul 31 2025

Formula

k : A014070(k) mod A006516(k) = binomial(2^k, k) mod binomial(2^n, 2) = 0.