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.

Showing 1-10 of 16 results. Next

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

Original entry on oeis.org

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

Views

Author

Joseph L. Pe, Jan 14 2002

Keywords

Comments

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

Examples

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

References

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

Crossrefs

Cf. A085793, A066726.

Programs

  • Mathematica
    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] &]
  • PARI
    is(n)=isprime(binomial(2n,n)+1) \\ Charles R Greathouse IV, May 15 2013

Extensions

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

A125221 Numbers k such that binomial(3k, k) + 1 is prime.

Original entry on oeis.org

0, 12, 43, 53, 120, 121, 184, 778, 823, 1179, 1212, 1670, 3187, 3353, 4282, 5420, 5592, 5826, 6526, 7405, 7581, 7619, 13213, 17258, 17532, 17644, 20274, 24404, 40406, 41981, 46955, 52968, 52968, 64289, 77776, 82571, 91231, 99641
Offset: 1

Views

Author

Alexander Adamchuk, Nov 24 2006

Keywords

Comments

a(39) > 10^5. - Robert Price, Aug 13 2024

Crossrefs

Cf. A125220, A066699, A066726.

Programs

  • Mathematica
    Do[f=Binomial[3n,n]+1;If[PrimeQ[f],Print[n]],{n,1,1000}]
  • PARI
    is(n)=isprime(binomial(3*n,n)+1) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

a(10)-a(15) from Robert G. Wilson v, Nov 26 2006
More terms from Stefan Steinerberger, Sep 07 2007
5 more terms from Ryan Propper, Jan 01 2008
a(24)-a(31) from Robert Price, Apr 17 2019
a(32)-a(38) from Robert Price, Aug 13 2024

A125220 Numbers k such that binomial(3k, k) - 1 is prime.

Original entry on oeis.org

1, 3, 7, 11, 49, 88, 93, 196, 216, 519, 655, 722, 858, 905, 991, 1654, 2277, 3275, 4214, 5047, 5924, 7359, 7953, 11188, 13286, 14626, 14687, 34365, 36014, 65613, 93663, 101805
Offset: 1

Views

Author

Alexander Adamchuk, Nov 25 2006

Keywords

Crossrefs

Cf. A125221 (binomial(3k, k) + 1 is prime).
Cf. A066699 (binomial(2k, k) + 1 is prime).
Cf. A066726 (binomial(2k, k) - 1 is prime).

Programs

  • Mathematica
    Do[f=Binomial[3n, n]-1; If[PrimeQ[f], Print[n]], {n, 1, 1000}]
Select[Range[4300],PrimeQ[Binomial[3#,#]-1]&] (* Harvey P. Dale, Aug 24 2017 *)
  • PARI
    is(n)=binomial(3*n,n)-1 \\ Charles R Greathouse IV, Feb 17 2017

Extensions

a(16)-a(19) from Robert G. Wilson v, Nov 26 2006
a(20)-a(29) from Robert Price, Apr 23 2019
a(30)-a(32) from Georg Grasegger, May 26 2025

A112861 Numbers k such that (2*k)!/(2*(k!)^2) - 1 is prime.

Original entry on oeis.org

2, 6, 10, 21, 45, 63, 306, 404, 437, 471, 646, 1174, 1192, 1334, 1975, 2397, 2410, 4305, 6111, 7852, 9488, 11120, 13304, 14408, 16075, 16238, 21188, 21659, 22025, 28673, 30793, 32178, 35278, 40049, 46516, 47836, 52157, 54531, 59897, 60275, 63362, 76139, 84219, 89024, 90783, 91605, 96761
Offset: 1

Views

Author

Hugo Pfoertner, Sep 30 2005

Keywords

Comments

a(48) > 100000. - Robert Price, Jul 25 2024

Crossrefs

Cf. A001700(n-1) = (2*n)!/(2*(n!)^2); A112862: primes of the form (2*n)!/(2*(n!)^2)-1; A112853: (2*n)!/n!-1 is prime; A112855: (2*n)!/n!+1 is prime; A066726: (2*n)!/(n!)^2-1 is prime; A066699: (2*n)!/(n!)^2+1 is prime; A112863: (2*n)!/(2*(n!)^2)+1 is prime.

Programs

  • Magma
    [n: n in [1..700] | IsPrime(Factorial(2*n) div (2*Factorial(n)^2)-1)]; // Vincenzo Librandi, Apr 10 2015
  • Mathematica
    Select[Range[10000], PrimeQ[(2 #)! / (2 (#!)^2) - 1 ] &] (* Vincenzo Librandi, Apr 10 2015 *)

Extensions

a(22)-a(26) from Vaclav Kotesovec, May 02 2021
a(27)-a(47) from Robert Price, Jul 25 2024

A112853 Numbers k such that (2*k)!/k!-1 is prime.

Original entry on oeis.org

2, 6, 14, 25, 114, 188, 193, 361, 712, 1767, 3195, 4995, 5682, 6485, 9937
Offset: 1

Views

Author

Hugo Pfoertner, Sep 25 2005

Keywords

Comments

Next term is > 9680.

Crossrefs

Cf. A112854 primes of the form (2*n)!/n!-1, A112855 (2*n)!/n!+1 is prime, A066726 (2*n)!/(n!)^2-1 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.

Programs

  • Mathematica
    Select[Range[7000], PrimeQ[(2#)!/#!-1]&] (* James C. McMahon, Jun 12 2024 *)

Extensions

a(15) from Michael S. Branicky, Jun 12 2024

A112855 Numbers n such that (2*n)!/n!+1 is prime.

Original entry on oeis.org

1, 2, 5, 10, 24, 30, 72, 340, 379, 712, 1020, 1647, 3654, 7923
Offset: 1

Views

Author

Hugo Pfoertner, Sep 25 2005

Keywords

Comments

Next term is > 10000.

Crossrefs

Cf. A112856 primes of the form (2*n)!/n!+1, A112853 (2*n)!/n!-1 is prime, A066726 (2*n)!/(n!)^2-1 is prime, A112859 (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.

Programs

  • Mathematica
    Select[Range[8000],PrimeQ[(2#)!/#!+1]&] (* Harvey P. Dale, Mar 23 2012 *)

A112863 Numbers k such that (2*k)!/(2*(k!)^2)+1 is prime.

Original entry on oeis.org

1, 3, 5, 6, 14, 19, 24, 45, 49, 62, 72, 122, 149, 197, 209, 222, 349, 409, 491, 527, 601, 675, 728, 769, 795, 853, 1039, 1318, 1599, 2069, 2279, 2567, 2634, 3286, 3482, 6880, 6919, 9117, 9276, 9564, 10898, 13061, 14570, 16493, 17366, 18773, 19901, 25569, 26611, 28322
Offset: 1

Views

Author

Hugo Pfoertner, Sep 30 2005

Keywords

Crossrefs

Cf. A001700, A112864, A112853, A112855, A066726, A112861.

Programs

  • Mathematica
    Select[Range[3000],PrimeQ[(2#)!/(2(#!)^2)+1]&] (* James C. McMahon, Jun 13 2024 *)
  • PARI
    isok(k) = ispseudoprime(1+binomial(2*k-1, k-1)); \\ Michel Marcus, Jun 14 2024

Extensions

a(42)-a(43) from Charles R Greathouse IV, Sep 26 2006
a(44)-a(50) from Michael S. Branicky, Jul 24 2024

A125240 Numbers k such that binomial(4k, k) - 1 is prime.

Original entry on oeis.org

1, 16, 36, 67, 95, 369, 383, 745, 1599, 2006, 2104, 2879, 3061, 9048, 9902, 12369, 15058, 18858, 21287, 22759, 24674, 33899, 43730, 55078, 86085
Offset: 1

Views

Author

Alexander Adamchuk, Nov 25 2006

Keywords

Comments

a(26) > 10^5 - Robert Price, Sep 15 2024

Crossrefs

Cf. A125241 = numbers n such that binomial(4n, n) + 1 is prime. Cf. A066699 = numbers n such that binomial(2n, n) + 1 is prime. Cf. A066726 = numbers n such that binomial(2n, n) - 1 is prime. Cf. A125220, A125221, A125242, A125243, A125244, A125245.

Programs

  • Mathematica
    Do[f=Binomial[4n, n]-1; If[PrimeQ[f], Print[n]], {n, 1, 1000}]

Extensions

More terms from Ryan Propper, Mar 28 2007
a(16)-a(23) from Robert Price, Apr 30 2019
a(24)-a(25) from Robert Price, Sep 15 2024

A125241 Numbers k such that binomial(4k, k) + 1 is prime.

Original entry on oeis.org

0, 1, 2, 6, 10, 11, 19, 28, 80, 123, 141, 147, 154, 198, 200, 346, 851, 887, 1038, 1329, 2045, 3228, 3274, 3588, 6794, 8045, 11911, 12184, 12327, 12515, 20089, 38173, 41026, 48914, 71772, 72130, 100726
Offset: 1

Views

Author

Alexander Adamchuk, Nov 25 2006

Keywords

Crossrefs

Cf. A125240 = numbers n such that binomial(4n, n) - 1 is prime. Cf. A066699 = numbers n such that binomial(2n, n) + 1 is prime. Cf. A066726 = numbers n such that binomial(2n, n) - 1 is prime. Cf. A125220, A125221, A125242, A125243, A125244, A125245.

Programs

  • Mathematica
    Do[f=Binomial[4n, n]+1; If[PrimeQ[f], Print[n]], {n, 1, 1000}]
Select[Range[8100],PrimeQ[Binomial[4#,#]+1]&] (* Harvey P. Dale, Aug 24 2014 *)

Extensions

More terms from Ryan Propper, Mar 28 2007
a(1)=0 added by Robert Price, May 01 2019
a(27)-a(34) from Robert Price, May 01 2019
a(35)-a(37) from Georg Grasegger, Jun 02 2025

A125242 Numbers k such that binomial(5k, k) - 1 is prime.

Original entry on oeis.org

5, 71, 111, 187, 239, 247, 473, 628, 847, 1478, 2687, 3530, 5175, 9113, 10968, 15039, 28929, 35649, 43481, 44455, 51269, 63975, 71723
Offset: 1

Views

Author

Alexander Adamchuk, Nov 25 2006

Keywords

Comments

a(24) > 10^5. Robert Price, Nov 16 2024

Crossrefs

Cf. A125243 = numbers n such that binomial(5n, n) + 1 is prime. Cf. A066699 = numbers n such that binomial(2n, n) + 1 is prime. Cf. A066726 = numbers n such that binomial(2n, n) - 1 is prime. Cf. A125220, A125221, A125240, A125241, A125244, A125245.

Programs

  • Mathematica
    Do[f=Binomial[5n, n]-1; If[PrimeQ[f], Print[n]], {n, 1, 1000}]
Select[Range[45000],PrimeQ[Binomial[5#,#]-1]&] (* Harvey P. Dale, Jan 02 2022 *)

Extensions

More terms from Ryan Propper, Jan 06 2007
a(15)-a(20) from Robert Price, May 03 2019
a(21)-a(23) from Robert Price, Nov 16 2024
Showing 1-10 of 16 results. Next

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