A093804 -id:A093804 - cp's OEIS frontend

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

N. J. A. Sloane

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

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

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.

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.

Cf. A002982 (n!-1 is prime), A064295. A088332 gives the primes.
Equals A090660 - 1.
Cf. A093804.

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

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 *)

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

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

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

A139159 a(n) = prime(n)! + 1.

Original entry on oeis.org

3, 7, 121, 5041, 39916801, 6227020801, 355687428096001, 121645100408832001, 25852016738884976640001, 8841761993739701954543616000001, 8222838654177922817725562880000001, 13763753091226345046315979581580902400000001, 33452526613163807108170062053440751665152000000001

Offset: 1

Artur Jasinski, Apr 11 2008

Cf. A039716.
Cf. A093804, A103319.
Cf. A139160, A139161, A139162, A139163, A139164, A139165, A139166, A139089, A139168.

Prime[Range[12]]!+1 (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

a(n)=prime(n)!+1 \\ Charles R Greathouse IV, Apr 29 2015

a(n) = A039716(n) + 1. - Michel Marcus, Nov 08 2013

More terms from Michel Marcus, Aug 10 2025

A103317 Primes p such that p! - 1 is also prime.

Original entry on oeis.org

3, 7, 379, 6917, 208003

Offset: 1

Jonathan Sondow, Jan 31 2005

The members are the primes in A002982 (n! - 1 is prime).

			3 is prime and 3! - 1 = 5 is prime, so 3 is a member.

R. K. Guy, Unsolved Problems in Number Theory, Section A2.

Eric Weisstein's World of Mathematics, Factorial Prime
Index entries for sequences related to factorial numbers.

Cf. A002982, A093804.

Select[Prime[Range[100]], PrimeQ[#! - 1] &] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

isok(n) = ispseudoprime(n) && ispseudoprime(n!-1); \\ Jinyuan Wang, Jan 20 2020

a(5)=208003 (found on Jul 27 2016, by Sou Fukui), added by Jinyuan Wang, Jan 20 2020

A163079 Primes p such that p$ + 1 is also prime. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

2, 3, 5, 31, 67, 139, 631, 9743, 16253, 17977, 27901, 37589

Offset: 1

Peter Luschny, Jul 21 2009

a(n) are the primes in A163077.

			5 is prime and 5$ + 1 = 30 + 1 = 31 is prime, so 5 is in the sequence.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A002981, A062363, A093804, A163077.

Cf. A056040. - Robert G. Wilson v, Aug 09 2010

a := proc(n) select(isprime,select(k -> isprime(A056040(k)+1),[$0..n])) end:

f[n_] := 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]; p = 2; lst = {}; While[p < 38000, a = f@p + 1; If[ PrimeQ@a, AppendTo[ lst, p]; Print@p]; p = NextPrime@p]; lst (* Robert G. Wilson v, Aug 08 2010 *)

is(k) = isprime(k) && ispseudoprime(1+k!/(k\2)!^2); \\ Jinyuan Wang, Mar 22 2020

a(8)-a(12) from Robert G. Wilson v, Aug 08 2010

A103319 Primes of the form p! + 1 where p is prime.

Original entry on oeis.org

3, 7, 39916801, 13763753091226345046315979581580902400000001, 33452526613163807108170062053440751665152000000001, 4470115461512684340891257138125051110076800700282905015819080092370422104067183317016903680000000000000001

Offset: 1

Jonathan Sondow, Jan 31 2005

The values of p are 2, 3, 11, 37, 41, 73 which is A093804 (with a different definition). Subsequence of A088332 (primes of the form n! + 1).

			2 and 2! + 1 = 3 are prime, so 3 is a member.

R. K. Guy, Unsolved Problems in Number Theory, Section A2.

R. 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, 2012 - From N. J. A. Sloane, Jun 13 2012
Eric Weisstein's World of Mathematics, Factorial Prime
Index entries for sequences related to factorial numbers.

Cf. A093804, A002981, A088332, A103318.

Select[Table[p!+1,{p,Prime[Range[30]]}],PrimeQ] (* Harvey P. Dale, Nov 28 2019 *)

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

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A139159 a(n) = prime(n)! + 1.

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A103317 Primes p such that p! - 1 is also prime.

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A163079 Primes p such that p$ + 1 is also prime. Here '$' denotes the swinging factorial function (A056040).

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A103319 Primes of the form p! + 1 where p is prime.

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