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.

Previous Showing 11-16 of 16 results.

A125243 Numbers k such that binomial(5k, k) + 1 is prime.

Original entry on oeis.org

0, 22, 86, 154, 160, 488, 705, 958, 975, 1262, 1932, 2845, 12718, 14434, 20337, 38834, 40433, 44874, 68279, 68724, 89911, 104765
Offset: 1

Views

Author

Alexander Adamchuk, Nov 25 2006

Keywords

Comments

a(22) > 100000. - Robert Price, Mar 20 2025

Crossrefs

Cf. A125242, A066699, A066726.
Cf. A125220, A125221, A125240, A125241, A125244, A125245.

Programs

  • Mathematica
    Do[f=Binomial[5n, n]+1; If[PrimeQ[f], Print[n]], {n, 1, 1000}]

Extensions

More terms from Ryan Propper, Jan 05 2007
a(1)=0 prepended and a(13)-a(18) added by Robert Price, May 11 2019
a(19)-a(21) from Robert Price, Mar 20 2025
a(22) from Georg Grasegger, May 06 2025

A125244 Numbers k such that binomial(6k, k) - 1 is prime.

Original entry on oeis.org

1, 7, 17, 22, 43, 343, 381, 461, 543, 923, 1045, 1182, 1486, 1839, 5643, 8260, 9009, 10947, 11793, 15915, 25151, 50923, 57095, 59977, 76513, 83383
Offset: 1

Views

Author

Alexander Adamchuk, Nov 25 2006

Keywords

Comments

a(27) > 100000. - Robert Price, May 09 2025

Crossrefs

Cf. A125245 = numbers n such that binomial(6n, 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, A125242, A125243.

Programs

  • Mathematica
    Do[f=Binomial[6n, n]-1; If[PrimeQ[f], Print[n]], {n, 1, 1000}]

Extensions

More terms from Ryan Propper, Mar 28 2007
a(18)-a(21) from Robert Price, May 13 2019
a(22)-a(23) from Georg Grasegger, May 08 2025
a(24)-a(26) from Robert Price, May 09 2025

A125245 Numbers k such that binomial(6k, k) + 1 is prime.

Original entry on oeis.org

0, 1, 2, 4, 10, 15, 98, 111, 118, 236, 280, 512, 1284, 1303, 1818, 2525, 2692, 4620, 8405, 11539, 13190, 21525, 30338, 48069, 50687, 56208, 56620, 81091, 101488
Offset: 1

Views

Author

Alexander Adamchuk, Nov 25 2006

Keywords

Comments

a(25) > 50000. - Robert Price, May 13 2019

Crossrefs

Cf. A125244 = numbers n such that binomial(6n, 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, A125242, A125243.

Programs

  • Mathematica
    Do[f=Binomial[6n, n]+1; If[PrimeQ[f], Print[n]], {n, 1, 1000}]

Extensions

More terms from Ryan Propper, Mar 28 2007
a(1)=0 and a(20)-a(24) from Robert Price, May 13 2019
a(25)-a(29) from Georg Grasegger, Jun 23 2025

A085793 Numbers k such that (k-1)*binomial(2k,k) + 1 is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 13, 17, 18, 22, 23, 28, 31, 48, 49, 52, 80, 99, 167, 201, 295, 372, 381, 391, 638, 653, 720, 779, 887, 1047, 1454, 1647, 1719, 2405, 3234, 3257, 3542, 3623, 3765, 3796, 4337, 4490, 5228, 6507, 8544, 9990, 10000, 12478, 13479, 15487, 17115
Offset: 1

Views

Author

Ed Pegg Jr, Jul 23 2003

Keywords

Comments

a(69) > 10^5 - Robert Price, Sep 15 2024

Examples

			9999 * 20000!/(10000!)^2 + 1 is prime

Links

Crossrefs

Cf. A066699, A066726.

Programs

Extensions

a(53)-a(68) from Robert Price, Sep 15 2024

A092751 Primes of the form (2*n)!/(n!)^2 - 1.

Original entry on oeis.org

5, 19, 251, 48619, 155117519, 30067266499541039, 6637553085023755473070799, 399608854866744452032002440111, 5717214010165655645594487649236004008072121335004636113518216597999
Offset: 1

Views

Author

Jorge Coveiro, Apr 12 2004

Keywords

Crossrefs

Cf. A000984, A066726.
Cf. A075840 = n such that (2*n)!/(n!)^2-1 is prime, A112860 = primes of the form (2*n)!/(n!)^2+1.

Programs

  • Mathematica
    Binomial[2#, # ] - 1 & /@ Select[ Range[150], PrimeQ[(2#)!/#!^2 - 1] &] (* Robert G. Wilson v, Apr 14 2004 *)

Extensions

Corrected and extended by Robert G. Wilson v, Apr 14 2004

A234963 Number of ways to write n = k + m with k > 0 and m > 2 such that C(2*sigma(k) + phi(m), sigma(k) + phi(m)/2) - 1 is prime, where sigma(k) is the sum of all positive divisors of k and phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 0, 3, 2, 2, 3, 3, 5, 3, 4, 3, 3, 3, 2, 3, 0, 3, 3, 4, 3, 0, 1, 2, 3, 1, 2, 3, 3, 1, 3, 3, 4, 1, 2, 3, 3, 2, 6, 4, 1, 4, 2, 3, 2, 2, 2, 4, 3, 2, 3, 3, 2, 4, 3, 3, 0, 2, 3, 1, 3, 1, 2, 0, 3, 1, 4, 4, 4, 1, 0, 5, 2, 1, 3, 2, 2, 1, 2, 1
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 01 2014

Keywords

Comments

Conjecture: a(n) > 0 for all n >= 180.
Clearly, this implies that there are infinitely many primes of the form C(2*n,n) - 1. We have verified the conjecture for n up to 10000.
Note that every n = 400, ..., 9123 can be written as k + m with k > 0 and m > 0 such that f(k, m) = sigma(k) + phi(m) is even and C(f(k, m) + 2, f(k, m)/2 + 1) + 1 is prime, but this fails for n = 9124.

Examples

			a(5) = 1 since 5 = 1 + 4 with C(2*sigma(1) + phi(4), sigma(1) + phi(4)/2) - 1 = C(4, 2) - 1 = 5 prime.
a(28) = 1 since 28 = 2 + 26 with C(2*sigma(2) + phi(26), sigma(2) + phi(26)/2) - 1 = C(18, 9) - 1 = 48619 prime.

Links

Crossrefs

Cf. A000010, A000040, A000203, A000984, A066726, A066699, A232270, A233544, A234451, A234470, A234503, A234514, A234567, A234615.

Programs

  • Mathematica
    sigma[n_] := DivisorSigma[1, n];
f[n_,k_] := Binomial[2*sigma[k] + EulerPhi[n-k], sigma[k] + EulerPhi[n-k]/2] - 1;
a[n_] := Sum[If[PrimeQ[f[n,k]], 1, 0], {k, 1, n-3}];
Table[a[n], {n, 1, 100}]
Previous Showing 11-16 of 16 results.

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