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A002981
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Numbers n such that n! + 1 is prime.
(Formerly M0908)
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61
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0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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 (f.firoozbakht(AT)sci.ui.ac.ir), 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 (jsondow(AT)alumni.princeton.edu)
150209 is also in the sequence, cf. the link to Caldwell's prime pages. - M. F. Hasler, Nov 04 2011
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REFERENCES
| C. K. Caldwell and Y. Gallot, On the primality of n!+-1 and 2*3*5*...*p+-1, Math. Comp., 71 (2001), 441-448.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 116, p. 40, Ellipses, Paris 2008.
H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3, 1987), 197-203.
R. K. Guy, Unsolved Problems in Number Theory, Section A2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| C. K. Caldwell, Primorial and Factorial Primes
C. K. Caldwell, 110059! + 1 on the Prime Pages
C. K. Caldwell, 150209! + 1 on the Prime Pages (Nov 03 2011).
N. Kuosa, Source for 6380.
H. Mishima, Factors of N!+1
R. 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
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EXAMPLE
| 3!+1 = 7 is prime, so 3 is in the sequence.
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MATHEMATICA
| v={0, 1, 2}; Do[If[ !PrimeQ[n+1]&&PrimeQ[n!+1], v=Append[v, n]; Print[v]], {n, 3, 29651}]
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PROG
| (PARI) for(n=0, 1e4, if(ispseudoprime(n!+1), print1(n", "))) \\ Charles R Greathouse IV, Jun 16 2011
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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: A041811 A056851 A048412 * A075641 A176316 A202211
Adjacent sequences: A002978 A002979 A002980 * A002982 A002983 A002984
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KEYWORD
| hard,nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| a(23) = 6380 sent in by Jud McCranie (JudMcCranie(AT)ugaalum.uga.edu), May 08 2000.
a(24) = 26951 from Ken Davis (kraden(AT)ozemail.com.au), May 24 2002
a(25) = 110059 found by PrimeGrid around Jun 11 2011, submitted by Eric W. Weisstein Jun 13 2011
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