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.
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. 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
CROSSREFS
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