A088054 Factorial primes: primes which are within 1 of a factorial number.

2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999

Offset: 1

Author

Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Nov 02 2003

Keywords

• easy
• nice
• nonn

Comments

Conjecture: 3 is the intersection of A002981 and A002982.

Examples

			3! + 1 = 7; 7! - 1 = 5039.
39916801 is a term because 11! + 1 is prime.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..29
• C. Caldwell's The Top Twenty, Factorial Primes.
• Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
• Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018. - From _N. J. A. Sloane_, Jun 13 2012
• Wikipedia, Factorial prime.

Crossrefs

Cf. A000142, A002981, A002982.

Union of A055490 and A088332.

Programs

Mathematica

t = {}; Do[ If[PrimeQ[n! - 1], AppendTo[t, n! - 1]]; If[PrimeQ[n! + 1], AppendTo[t, n! + 1]], {n, 50}]; t (* Robert G. Wilson v *)
Union[Select[Range[50]!-1, PrimeQ], Select[Range[50]!+1, PrimeQ]] (Noe)
fp[n_] := Module[{nf=n!}, Select[{nf-1,nf+1},PrimeQ]]; Flatten[ Table[ fp[i],{i,50}]] (* Harvey P. Dale, Dec 18 2010 *)
Select[Flatten[#+{-1,1}&/@(Range[50]!)],PrimeQ] (* Harvey P. Dale, Apr 08 2019 *)

Python

from itertools import count, islice
from sympy import isprime
def A088054_gen(): # generator of terms
    f = 1
    for k in count(1):
        f *= k
        if isprime(f-1):
            yield f-1
        if isprime(f+1):
            yield f+1
A088054_list = list(islice(A088054_gen(),10)) # Chai Wah Wu, Feb 18 2022

Extensions

Corrected by Paul Muljadi, Oct 11 2005

More terms from Robert G. Wilson v and T. D. Noe, Oct 12 2005