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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 (list; graph; refs; listen; history; text; internal format)
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_{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

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).

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), 441-448.

N. Kuosa, Source for 6380.

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, 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

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Last modified August 30 02:28 EDT 2014. Contains 246215 sequences.