A136327 Numbers k > 1 such that binomial(2k-1, k-1) == 1 (mod k).

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251

Offset: 1

Author

Franz Vrabec, Mar 26 2008

Keywords

• nonn

Comments

k such that A099905(k) = 1.

Contains primes, squares of odd primes and cubes of primes >= 5.

See A228562 for terms that are neither primes nor prime powers. [Joerg Arndt, Aug 27 2013]

Examples

			a(3) = 5 because binomial(9, 4) = 126 == 1 (mod 5).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• McIntosh, R. J. (1995), On the converse of Wolstenholme's theorem, Acta Arithm., LXXI.4 (1995), 381-389.

Crossrefs

Cf. A099905.

Programs

Maple

filter:= k -> binomial(2*k-1,k-1) mod k = 1:
select(filter, [$1..1000]); # Robert Israel, Feb 11 2025

Mathematica

Select[Range[300], Mod[Binomial[2# - 1, # - 1], #] == 1 &] (* Alonso del Arte, May 11 2014 *)

PARI

isok(n) = (binomial(2*n-1, n-1) % n) == 1; \\ Michel Marcus, Aug 26 2013

Extensions

Name corrected by Robert Israel, Feb 11 2025