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 1-7 of 7 results.

A118812 Primes of the form (2n)! - n! + 1.

Original entry on oeis.org

2, 23, 87178286161, 403291461126605629356979201, 5502622159812088949850305428800254867109635014075023360001
Offset: 1

Views

Author

Paolo P. Lava and Giorgio Balzarotti, May 23 2006

Keywords

Comments

a(6) is a 41025-digit prime corresponding to n = 5666, and a(7) is a 288639-digit prime corresponding to n = 32918. See A237443 for additional values of n. - Kellen Shenton, Dec 21 2024
Primes in sequence A237580 = n -> (2n)! - n! + 1, i.e., the terms of that sequence which coincide with A237579(n) = least prime factor of (2n)! - n! + 1. - M. F. Hasler, Feb 09 2014

Examples

			For n=2, (2*2)! - 2! + 1 = 24 - 2 + 1 = 23, which is prime.

References

  • G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 159.

Crossrefs

Cf. A237443 (corresponding values of n).

Programs

  • Maple
    PFACT:=proc(N) local i,r; for i from 1 by 1 to N do r:=(2*i)!-i!+1; if isprime(r) then print(i); fi; od; end: PFACT(100);
  • Mathematica
    Select[Table[(2n)!-n!+1,{n,30}],PrimeQ] (* Harvey P. Dale, May 05 2018 *)
  • PARI
    for(n=1,999,ispseudoprime(p=(2*n)!-n!+1)&&print1(p",")) \\ M. F. Hasler, Feb 09 2014

Formula

A118812 = A237580 o A237443 = A237579 o A237443. - M. F. Hasler, Feb 09 2014

A237579 Least prime factor of (2n)! - n! + 1 (= A237580(n)).

Original entry on oeis.org

2, 23, 5, 59, 7, 59, 87178286161, 29, 11, 443, 13, 587, 403291461126605629356979201, 3307, 17, 43, 19, 131, 83, 2791, 23, 113, 5502622159812088949850305428800254867109635014075023360001, 659, 761, 6108689, 29, 233, 31, 67, 181, 25649409970727, 1561016461, 151, 37, 223, 53, 139, 41, 29209, 43, 61
Offset: 1

Views

Author

M. F. Hasler, Feb 09 2014

Keywords

Comments

For n=0 the expression equals 1 and has no prime factor at all.
The larger terms, and/or those ending in ...01, correspond to indices (listed in A237443) for which the expression itself is a prime (listed in A118812).

Links

Crossrefs

Cf. A237580, A237443, A118812.

Programs

  • PARI
    n -> factor((2*n)!-n!+1)[1,1]

Extensions

a(26)-a(42) from Chai Wah Wu, Oct 15 2019
Typo in a(32) corrected by Seth A. Troisi, May 13 2022

A237580 a(n) = (2n)! - n! + 1.

Original entry on oeis.org

1, 2, 23, 715, 40297, 3628681, 479000881, 87178286161, 20922789847681, 6402373705365121, 2432902008173011201, 1124000727777567763201, 620448401733238960358401, 403291461126605629356979201, 304888344611713860414325708801, 265252859812191058635000805632001
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2014

Keywords

Comments

Primes are listed in A118812, the corresponding indices in A237443; the least prime factor of a(n) in A237579.

Links

Crossrefs

Cf. A000142 (n!), A010050 ((2n)!)

Programs

A242487 Numbers k such that (2*k)! + k! + 1 is prime.

Original entry on oeis.org

0, 3, 8, 13, 19, 423, 585, 2746, 2855
Offset: 1

Views

Author

Seiichi Manyama, Mar 22 2018

Keywords

Comments

a(10) > 10000. - Michael S. Branicky, May 03 2025

Examples

			0! + 0! + 1 =   3 is prime.
6! + 3! + 1 = 727 is prime.

Crossrefs

A300947 gives the primes.
Cf. A002981, A237443.

Programs

  • Maple
    select(k->isprime(factorial(2*k)+factorial(k)+1),[$0..600]); # Muniru A Asiru, May 27 2018
  • Mathematica
    Flatten[{0, Select[Range[100], PrimeQ[(2*#)! + #! + 1] &]}] (* Vaclav Kotesovec, Mar 25 2018 *)
  • PARI
    isok(k) = ispseudoprime((2*k)!+k!+1); \\ Altug Alkan, Mar 22 2018

Extensions

a(8)-a(9) from Michael S. Branicky, Apr 16 2023

A118813 Primes of the form (2n)! - n! - 1.

Original entry on oeis.org

3628679, 87178286159, 20922789847679, 265252859812191058635000805631999
Offset: 1

Views

Author

Paolo P. Lava and Giorgio Balzarotti, May 23 2006

Keywords

Comments

The term a(7) (2407 digits) is too large to include even in the b-file.
All the numbers end in 9.
For all n, (2n)! - n! - 1 has exactly A027868(n) trailing 9s. - Rick L. Shepherd, Feb 16 2014
Generated by n = 5, 7, 8, 15, 103, 179, .... - R. J. Mathar, Sep 20 2012 (now A238011)

Examples

			For n=5, (2*5)! - 5! - 1 = 3628800 - 120 - 1 = 3628679, which is prime.

References

  • G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 159.

Links

Crossrefs

Cf. A238011 (corresponding n), A118812, A237443.

Programs

  • Maple
    PFACT:=proc(N) local i,r; for i from 1 by 1 to N do r:=(2*i)!-i!-1; if isprime(r) then print(i); fi; od; end: PFACT(100);

A238011 Numbers k such that (2k)! - k! - 1 is prime.

Original entry on oeis.org

5, 7, 8, 15, 103, 179, 473, 2054, 3595, 4039
Offset: 1

Views

Author

Rick L. Shepherd, Feb 16 2014

Keywords

Comments

No more terms up to 5000. The first six terms are as previously stated in A118813. All terms found and primes proved by OpenPFGW.
No more terms up to 10000. - Michael S. Branicky, Dec 23 2024

Examples

			5 is a term because (2*5)! - 5! - 1 = 3628679 is prime.
4039 is a term because (2*4039)! - 4039! - 1 is prime (28058 digits).

Links

Crossrefs

Cf. A118813 (corresponding primes), A237443, A118812.

A379380 Prime numbers p such that (2*p)# - p# + 1 is prime, where p# = A034386(p).

Original entry on oeis.org

1, 2, 5, 17, 103, 113, 193, 277, 3671, 9419, 12671, 30809, 34439, 83833, 89227, 106103, 107101, 175303, 398903
Offset: 1

Views

Author

Kellen Shenton, Dec 21 2024

Keywords

Examples

			For p = 17: the product of the primes from 2 to p*2 is 2*3*5*7*11*13*17*19*23*29*31 = 200560490130, and the product of the primes from 2 to p is 2*3*5*7*11*13*17 = 510510.  200560490130 - 510510 + 1 = 200559979621, a prime number.

Crossrefs

Cf. A034386, A237443.
Showing 1-7 of 7 results.

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