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

0,1

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.

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.

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

Example

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.

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 *)

Prog

(PARI) a(n)=nextprime(n!); \\ _R. J. (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

Crossrefs

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

Sequence in context: A158927 A367857 A121258 * A309574 A132697 A160689

Adjacent sequences: A087418 A087419 A087420 * A087422 A087423 A087424

Keyword

nonn

Author

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

Extensions

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

Status

approved