

A002981


Numbers n such that n! + 1 is prime.
(Formerly M0908)


66



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

1,3


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_{dn} d! is prime) since Sum_{dn} 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


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), 197203.
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).


LINKS

Table of n, a(n) for n=1..22.
Chris K. Caldwell, Primorial and Factorial Primes
Chris K. Caldwell, 110059! + 1 on the Prime Pages
Chris K. Caldwell, 150209! + 1 on the Prime Pages (Nov 03 2011).
Chris K. Caldwell and Y. Gallot, On the primality of n!+1 and 2*3*5*...*p+1, Math. Comp., 71 (2001), 441448.
N. Kuosa, Source for 6380.
Romeo Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC2012) and another new proof, Arxiv preprint arXiv:1202.3670, 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
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


EXAMPLE

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


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 *)


PROG

(PARI) for(n=0, 1e4, if(ispseudoprime(n!+1), print1(n", "))) \\ Charles R Greathouse IV, Jun 16 2011


CROSSREFS

Cf. A002982 (n!1 is prime), A064295. A088332 gives the primes.
Equals A090660  1.
See also A002982 (n!1 is prime).
Cf. A093804.
Sequence in context: A229066 A056851 A048412 * A232212 A232219 A235474
Adjacent sequences: A002978 A002979 A002980 * A002982 A002983 A002984


KEYWORD

hard,more,nonn,nice


AUTHOR

N. J. A. Sloane.


EXTENSIONS

Term 6380 sent in by Jud McCranie, May 08 2000.
Term 26951 from Ken Davis (kraden(AT)ozemail.com.au), May 24 2002
Term 110059 found by PrimeGrid around Jun 11 2011, submitted by Eric W. Weisstein, Jun 13 2011
Term 150209 by Rene Dohmen, Jun 09 2012.


STATUS

approved



