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).

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, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199

Offset: 1

Author

Vladimir Shevelev, May 08 2008

Keywords

• nonn

Comments

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.

Numbers n such that A178105(n) = 0. - Michel Marcus, Feb 07 2016

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..10000
• V. Shevelev, On divisibility of binomial(n-i-1,i-1) by i, Intl. J. of Number Theory, 3, no.1 (2007), 119-139.

Crossrefs

Cf. A000040, A001248, A077800, A037074, A178105.

Programs

Mathematica

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 *)

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