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A002981 Numbers k such that k! + 1 is prime.
(Formerly M0908)
112
0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, 288465 (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
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, 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 & N. J. A. Sloane, Correspondence, 1991
R. K. Guy & N. J. A. Sloane, Correspondence, 1985
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
Eric Weisstein's World of Mathematics, Factorial Prime
Eric Weisstein's World of Mathematics, Integer Sequence Primes
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, 500, if(ispseudoprime(n!+1), print1(n", "))) \\ Charles R Greathouse IV, Jun 16 2011
(Magma) [n: n in [0..800] | IsPrime(Factorial(n)+1)]; // Vincenzo Librandi, Oct 31 2018
(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
CROSSREFS
Cf. A002982 (n!-1 is prime), A064295. A088332 gives the primes.
Equals A090660 - 1.
Cf. A093804.
Sequence in context: A284046 A048412 A259428 * A294637 A295613 A232212
KEYWORD
nonn,nice,hard,more
AUTHOR
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
STATUS
approved

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Last modified March 18 21:02 EDT 2024. Contains 370951 sequences. (Running on oeis4.)