A002981 Numbers k such that k! + 1 is prime.

0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, 288465, 308084, 422429

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard
• more

Comments

If n + 1 is prime then (by Wilson's theorem) n + 1 divides n! + 1. Thus for n > 2 if n + 1 is prime n is not in the sequence. - Farideh Firoozbakht, Aug 22 2003

For n > 2, n! + 1 is prime <==> nextprime((n+1)!) > (n+1)nextprime(n!) and we can conjecture that for n > 2 if n! + 1 is prime then (n+1)! + 1 is not prime. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 03 2004

The prime members are in A093804 (numbers n such that Sum_{d|n} d! is prime) since Sum_{d|n} d! = n! + 1 if n is prime. - Jonathan Sondow

150209 is also in the sequence, cf. the link to Caldwell's prime pages. - M. F. Hasler, Nov 04 2011

Examples

			3! + 1 = 7 is prime, so 3 is in the sequence.
		

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 116, p. 40, Ellipses, Paris 2008.
• Harvey Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3, 1987), 197-203.
• Richard K. Guy, Unsolved Problems in Number Theory, Section A2.
• F. Le Lionnais, Les Nombres Remarquables, Paris, Hermann, 1983, p. 100.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 118.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 70.

Links

• A. Borning, Some results for k!+-1 and 2.3.5...p+-1, Math. Comp., 26 (1972), 567-570.
• Chris K. Caldwell, Factorial Primes.
• Chris K. Caldwell, 110059! + 1 on the Prime Pages.
• Chris K. Caldwell, 150209! + 1 on the Prime Pages (Oct 31, 2011).
• Chris K. Caldwell, 288465! + 1 on the Prime Pages (Jan 12, 2022).
• Chris K. Caldwell and Y. Gallot, On the primality of n!+-1 and 2*3*5*...*p+-1, Math. Comp., 71 (2001), 441-448.
• 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.
• H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3, 1987), 197-203. (Annotated scanned copy)
• H. Dubner and N. J. A. Sloane, Correspondence, 1991.
• Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
• R. K. Guy and N. J. A. Sloane, Correspondence, 1985.
• N. Kuosa, Source for 6380.
• Des MacHale and Joseph Manning, Maximal runs of strictly composite integers, The Mathematical Gazette, 99, pp 213-219 (2015).
• Romeo Mestrovic, 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. - From N. J. A. Sloane, Jun 13 2012
• Hisanori Mishima, Factors of N!+1.
• Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations.
• Titus Piezas III, 2004. Solving Solvable Sextics Using Polynomial Decomposition.
• PrimePages, Factorial Primes.
• Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
• Apoloniusz Tyszka, A conjecture which implies that there are infinitely many primes of the form n!+1, Preprint, 2017.
• Apoloniusz Tyszka, A common approach to the problem of the infinitude of twin primes, primes of the form n! + 1, and primes of the form n! - 1, 2018.
• Apoloniusz Tyszka, On sets X subset of N for which we know an algorithm that computes a threshold number t(X) in N such that X is infinite if and only if X contains an element greater than t(X), 2019.
• Apoloniusz Tyszka, On sets X, subset of N, whose finiteness implies that we know an algorithm which for every n, element of N, decides the inequality max (X) < n, (2019).
• Eric Weisstein's World of Mathematics, Factorial Prime.
• Eric Weisstein's World of Mathematics, Integer Sequence Primes.
• Index entries for sequences related to factorial numbers.

Crossrefs

Cf. A002982 (n!-1 is prime), A064295. A088332 gives the primes.

Equals A090660 - 1.

Cf. A093804.

Programs

Magma

[n: n in [0..800] | IsPrime(Factorial(n)+1)]; // Vincenzo Librandi, Oct 31 2018

Mathematica

v = {0, 1, 2}; Do[If[ !PrimeQ[n + 1] && PrimeQ[n! + 1], v = Append[v, n]; Print[v]], {n, 3, 29651}]
Select[Range[100], PrimeQ[#! + 1] &] (* Alonso del Arte, Jul 24 2014 *)

PARI

for(n=0,500,if(ispseudoprime(n!+1),print1(n", "))) \\ Charles R Greathouse IV, Jun 16 2011

Python

from sympy import factorial, isprime
for n in range(0,800):
    if isprime(factorial(n)+1):
        print(n, end=', ') # Stefano Spezia, Jan 10 2019

Extensions

a(19) sent in by Jud McCranie, May 08 2000

a(20) from Ken Davis (kraden(AT)ozemail.com.au), May 24 2002

a(21) found by PrimeGrid around Jun 11 2011, submitted by Eric W. Weisstein, Jun 13 2011

a(22) from Rene Dohmen, Jun 09 2012

a(23) from Rene Dohmen, Jan 12 2022

a(24)-a(25) from Dmitry Kamenetsky, Jun 19 2024