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A082916 Numbers k such that k and binomial(2*k, k) are relatively prime.

%I A082916 #27 Jun 28 2025 11:15:02
%S A082916 0,1,3,5,7,9,11,13,17,19,23,25,27,29,31,37,39,41,43,47,49,53,55,59,61,
%T A082916 67,71,73,79,81,83,89,93,97,101,103,107,109,111,113,119,121,125,127,
%U A082916 131,137,139,149,151,155,157,161,163,167,169,173,179,181,185,191,193,197
%N A082916 Numbers k such that k and binomial(2*k, k) are relatively prime.
%C A082916 Also the numbers k such that every base-p digit of k is less than p/2, for every prime divisor p of k. Contains all odd primes and their powers. - _David Radcliffe_, Jun 28 2025
%D A082916 J. Glaisher, On the residue of a binomial-theorem coefficient with respect to a prime modulus, Quarterly J. of Pure and Applied Math. 30 (1899), 150-156.
%H A082916 Chai Wah Wu, <a href="/A082916/b082916.txt">Table of n, a(n) for n = 1..10000</a>
%F A082916 It seems that a(n) is asymptotic to c*n*log(n) with 0.7<c<0.8.
%t A082916 Select[Range[0, 100], CoprimeQ[Binomial[2*#, #], #] &] (* _Amiram Eldar_, May 24 2020 *)
%o A082916 (PARI) isok(n) = gcd(n, binomial(2*n, n)) == 1; \\ _Michel Marcus_, Dec 04 2013
%o A082916 (Python)
%o A082916 from math import gcd
%o A082916 A082916_list, b = [], 1
%o A082916 for n in range(10**5):
%o A082916     if gcd(n,b) == 1:
%o A082916         A082916_list.append(n)
%o A082916     b = b*(4*n+2)//(n+1) # _Chai Wah Wu_, Mar 25 2016
%Y A082916 Cf. A000984 (central binomial coefficients). Contains A061345.
%K A082916 nonn
%O A082916 1,3
%A A082916 _Benoit Cloitre_, May 25 2003