A085793 -id:A085793 - cp's OEIS frontend

A066699 Numbers k such that binomial(2k,k)+1 is prime.

1, 2, 4, 7, 12, 19, 22, 38, 46, 62, 68, 72, 84, 166, 184, 214, 340, 348, 445, 517, 692, 817, 1316, 1381, 2554, 2713, 5261, 6209, 6735, 7920, 8207, 8772, 9530, 13075, 13302, 13405, 15002, 16371, 19346, 24151, 26555, 28188, 29235, 33536, 43338, 44048, 65576, 65930, 68666, 78285

Offset: 1

Joseph L. Pe, Jan 14 2002

nonn

a(45) > 40000. All the primes corresponding to terms up to a(44) have been certified by the PFGW software performing the Brillhart-Lehmer-Selfridge N-1 test. - Giovanni Resta, Apr 05 2017

a(51) > 100000. - Robert Price, Jul 02 2024

			C(4,2) + 1 = 7, a prime; so 2 is a term of the sequence.

Aigner and Ziegler. Proofs from the Book, 2nd edition. Springer-Verlag, 2001.

Cf. A085793, A066726.

Do[If[PrimeQ[Binomial[2 a, a]+1], a >>>"C:\prime.txt"],{a,1,20000}] (* Ed Pegg Jr *)
Select[Range[1, 5 * 10^2], PrimeQ[Binomial[2* #, # ] + 1] &]

is(n)=isprime(binomial(2n,n)+1) \\ Charles R Greathouse IV, May 15 2013

More terms (not certified primes) from Jason Earls and Robert G. Wilson v, Jan 15 2002
More terms from Ed Pegg Jr, Sep 10 2003
a(40)-a(44) from Giovanni Resta, Apr 05 2017
a(45)-a(50) from Robert Price, Jul 02 2024

A066726 Numbers n such that binomial(2n, n) - 1 is prime.

Original entry on oeis.org

2, 3, 5, 9, 15, 29, 43, 51, 113, 184, 213, 222, 267, 279, 369, 402, 441, 603, 812, 839, 902, 1422, 1542, 1824, 2983, 3065, 3911, 3958, 4192, 4587, 4865, 5543, 5837, 7902, 9299, 9722, 10412, 10648, 11498, 12803, 14428, 15876, 20173, 26311, 38927, 52210, 54189, 59757, 60454, 72094, 76899, 85033, 91059, 91059

Offset: 1

Robert G. Wilson v, Jan 15 2002

nonn

I.e., numbers n such that (2*n)!/(n!)^2-1 is prime. - Hugo Pfoertner, Sep 25 2005
The next term is > 30000. - Vaclav Kotesovec, May 03 2021
a(55) > 100000. - Robert Price, Jul 02 2024

Cf. A066699, A085793.

Cf. A092751 = primes of the form (2*n)!/(n!)^2-1, A112853 = (2*n)!/n!-1 is prime, A112855 = (2*n)!/n!+1 is prime, A066699 = (2*n)!/(n!)^2+1 is prime, A112861 = (2*n)!/(2*(n!)^2)-1 is prime, A112863 = (2*n)!/(2*(n!)^2)+1 is prime. - Hugo Pfoertner, Sep 25 2005

Do[ If[ PrimeQ[ Binomial[2n, n] - 1], Print[n]], {n, 1, 2000} ]

is(n)=isprime(binomial(2*n,n)-1) \\ Charles R Greathouse IV, Feb 17 2017

More terms from Ed Pegg Jr, Sep 10 2003
Edited by N. J. A. Sloane, Aug 23 2008 at the suggestion of R. J. Mathar
a(43)-a(44) from Vaclav Kotesovec, May 03 2021
a(45)-a(54) from Robert Price, Jul 02 2024

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A066699 Numbers k such that binomial(2k,k)+1 is prime.

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