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A138389 Binomial primes: positive integers n such that every i not coprime to n and not exceeding n/2 does not divide binomial(n-i-1,i-1).

%I A138389 #23 Feb 07 2016 10:30:38
%S A138389 1,2,3,4,5,6,7,8,9,10,11,12,13,15,17,19,20,21,23,24,25,29,31,33,35,37,
%T A138389 41,43,47,49,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,121,
%U A138389 127,131,137,139,143,149,151,157,163,167,169,173,179,181,191,193,197,199
%N A138389 Binomial primes: positive integers n such that every i not coprime to n and not exceeding n/2 does not divide binomial(n-i-1,i-1).
%C A138389 Note that every i not exceeding n/2 for which (n,i)=1 divides binomial(n-i-1,i-1). For n>33, a(n) is either prime or square of a prime or a product of twin primes. For a proof, see link of V. Shevelev.
%C A138389 Numbers n such that A178105(n) = 0. - _Michel Marcus_, Feb 07 2016
%H A138389 Peter J. C. Moses, <a href="/A138389/b138389.txt">Table of n, a(n) for n = 1..10000</a>
%H A138389 V. Shevelev, <a href="http://dx.doi.org/10.1142/S179304210700078X">On divisibility of binomial(n-i-1,i-1) by i</a>, Intl. J. of Number Theory, 3, no.1 (2007), 119-139.
%t A138389 Select[Range@ 200, Function[n, NoneTrue[Select[Range@ Floor[n/2], ! CoprimeQ[#, n] &], Divisible[Binomial[n - # - 1, # - 1], #] &]]] (* _Michael De Vlieger_, Feb 07 2016, Version 10 *)
%o A138389 (PARI) isok(n) = {my(md = -1); for (d=2, n\2, if (((binomial(n-d-1,d-1) % d) == 0) && (gcd(n, d) > 1), if (md == -1, md = d, md = min(d, md)));); (md == -1);} \\ _Michel Marcus_, Feb 07 2016
%Y A138389 Cf. A000040, A001248, A077800, A037074, A178105.
%K A138389 nonn
%O A138389 1,2
%A A138389 _Vladimir Shevelev_, May 08 2008