cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Robert G. Wilson v, Jan 15 2002

Keywords

Comments

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

Crossrefs

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

Programs

  • Mathematica
    Do[ If[ PrimeQ[ Binomial[2n, n] - 1], Print[n]], {n, 1, 2000} ]
  • PARI
    is(n)=isprime(binomial(2*n,n)-1) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

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

A075840 Primes of the form (2*n)!/(n!)^2+1.

Original entry on oeis.org

2, 3, 7, 71, 3433, 2704157, 35345263801, 2104098963721, 6892620648693261354601, 410795449442059149332177041, 1520803477811874490019821888415218657, 5949105755928259715106809205795376486501, 1480212998448786189993816895482588794876101
Offset: 1

Views

Author

Donald S. McDonald, Oct 14 2002

Keywords

Examples

			7 is a term because C(4,2)+1 = 6+1 = 7 is prime.

References

  • New Zealand Science Monthly, Bulletin Board, Feb. 1999. Binomial(300,150)+185 = nextprime.

Links

Crossrefs

Cf. A092751 = n such that (2*n)!/(n!)^2+1 is prime, A112858 = primes of the form (2*n)!/(n!)^2-1.
Cf. A000984, n's are in A066699.

Programs

  • Magma
    [a: n in [0..100] | IsPrime(a) where a is Factorial(2*n) div Factorial(n)^2+1]; // Vincenzo Librandi Mar 17 2015
  • Mathematica
    a = Select[ Range[100], PrimeQ[Binomial[2#, # ] + 1] & ]; Binomial[2a, a] + 1
Select[Table[(2 n)! / (n!)^2 + 1, {n, 0, 80}], PrimeQ] (* Vincenzo Librandi, Mar 17 2015 *)
  • PARI
    v=[]; for(n=0,100,x=bin(2*n,n)+1; if(isprime(x), v=concat(v,x),)); v

Extensions

Edited by Robert G. Wilson v, Oct 15 2002
Definition corrected by Alexander Adamchuk, Nov 30 2007
Edited by N. J. A. Sloane, Nov 30 2007
a(13) from Vincenzo Librandi, Mar 17 2015
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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