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

%I A042943 #22 Aug 01 2025 01:55:52
%S A042943 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,
%T A042943 45,46,47,49,50,51,53,55,58,59,61,62,65,66,67,69,70,71,73,74,75,77,79,
%U A042943 81,83,85,86,87,89,91,93,94,95,97,98,99,101,102,103,106,107,109,111
%N A042943 Numbers k such that binomial(2^k, k) is divisible by binomial(2^k, 2).
%C A042943 Does not contain multiples of 4 (A008586).
%F A042943 k : A014070(k) mod A006516(k) = binomial(2^k, k) mod binomial(2^n, 2) = 0.
%t A042943 Select[Range[150],Divisible[Binomial[2^#,#],Binomial[2^#,2]]&]  (* _Harvey P. Dale_, Mar 24 2011 *)
%o A042943 (PARI) isok(k) = (binomial(2^k, k) % binomial(2^k, 2)) == 0; \\ _Michel Marcus_, May 14 2018
%o A042943 (Python)
%o A042943 from math import comb
%o A042943 from itertools import count, islice
%o A042943 def A042943_gen(startvalue=1): # generator of terms >= startvalue
%o A042943     for k in count(max(startvalue,1)):
%o A042943         if comb(m:=1<<k,k)%comb(m,2) == 0:
%o A042943             yield k
%o A042943 A042943_list = list(islice(A042943_gen(),30)) # _Chai Wah Wu_, Jul 31 2025
%Y A042943 Cf. A014070, A006516.
%K A042943 easy,nonn
%O A042943 1,2
%A A042943 _Labos Elemer_, Apr 11 2001