A087421 Smallest prime >= n!.

2, 2, 2, 7, 29, 127, 727, 5051, 40343, 362897, 3628811, 39916801, 479001629, 6227020867, 87178291219, 1307674368043, 20922789888023, 355687428096031, 6402373705728037, 121645100408832089, 2432902008176640029, 51090942171709440031, 1124000727777607680031

Offset: 0

Author

Mitch Cervinka (puritan(AT)planetkc.com), Oct 22 2003

Keywords

• nonn

Comments

n! is prime only when n=2. When n>2, for n!+m to be prime, m must be relatively prime to all the numbers from 2 to n. In particular, if m is between 2 and n, then (n!+m) will be divisible by m. Thus a(n) must be either n!+1, or else larger than n!+n.

Examples

			a(0) = 2 since 0! = 1 and 2 is the smallest prime >= 1.
a(4) = 29 since 4! = 24 and 29 is the smallest prime >= 24.
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 0..400
• Antonín Čejchan, Michal Křížek, and Lawrence Somer, On Remarkable Properties of Primes Near Factorials and Primorials, Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.4.

Crossrefs

Cf. A000142, A006990, A007918, A033932, A037151.

Programs

Mathematica

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Table[ NextPrim[n! - 1], {n, 0, 20}] (* Robert G. Wilson v, Oct 25 2003 *)
Join[{2,2,2},NextPrime[Range[3,25]!]]  (* Harvey P. Dale, Feb 23 2011 *)

PARI

a(n)=nextprime(n!); \\ R. J. Cano, Apr 08 2018

Python

from sympy import factorial, nextprime
def a(n): return nextprime(factorial(n)-1)
print([a(n) for n in range(23)]) # Michael S. Branicky, May 22 2022

Formula

a(n) = min { p[i] | p[i]>=n! }, where p[i] is the set of prime numbers.

a(n) = A007918(A000142(n)). - Michel Marcus, Apr 09 2018

Extensions

Edited, corrected and extended by Robert G. Wilson v and Ray Chandler, Oct 25 2003