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A002981 Numbers k such that k! + 1 is prime.
(Formerly M0908)

%I M0908

%S 0,1,2,3,11,27,37,41,73,77,116,154,320,340,399,427,872,1477,6380,

%T 26951,110059,150209

%N Numbers k such that k! + 1 is prime.

%C 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

%C 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

%C 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_

%C 150209 is also in the sequence, cf. the link to Caldwell's prime pages. - _M. F. Hasler_, Nov 04 2011

%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 116, p. 40, Ellipses, Paris 2008.

%D Harvey Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3, 1987), 197-203.

%D Richard K. Guy, Unsolved Problems in Number Theory, Section A2.

%D F. Le Lionnais, Les Nombres Remarquables, Paris, Hermann, 1983, p. 100.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H A. Borning, <a href="http://dx.doi.org/10.1090/S0025-5718-1972-0308018-5 ">Some results for k!+-1 and 2.3.5...p+-1</a>, Math. Comp., 26 (1972), 567-570.

%H Chris K. Caldwell, <a href="http://primes.utm.edu/top20/page.php?id=30">Factorial Primes</a>

%H Chris K. Caldwell, <a href="http://primes.utm.edu/primes/page.php?id=100445">110059! + 1 on the Prime Pages</a>

%H Chris K. Caldwell, <a href="http://primes.utm.edu/primes/page.php?id=102627">150209! + 1 on the Prime Pages</a> (Nov 03 2011).

%H Chris K. Caldwell and Y. Gallot, <a href="http://dx.doi.org/10.1090/S0025-5718-01-01315-1">On the primality of n!+-1 and 2*3*5*...*p+-1</a>, Math. Comp., 71 (2001), 441-448.

%H H. Dubner, <a href="/A006794/a006794.pdf">Factorial and primorial primes</a>, J. Rec. Math., 19 (No. 3, 1987), 197-203. (Annotated scanned copy)

%H H. Dubner & N. J. A. Sloane, <a href="/A002981/a002981.pdf">Correspondence, 1991</a>

%H R. K. Guy & N. J. A. Sloane, <a href="/A005648/a005648.pdf">Correspondence, 1985</a>

%H N. Kuosa, <a href="http://www.hut.fi/~nkuosa/primeform/">Source for 6380.</a>

%H Des MacHale and Joseph Manning, <a href="http://dx.doi.org/10.1017/mag.2015.28">Maximal runs of strictly composite integers</a>, The Mathematical Gazette, 99, pp 213-219 (2015).

%H Romeo Mestrovic, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - From N. J. A. Sloane, Jun 13 2012

%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha104.htm">Factors of N!+1</a>

%H Rudolf Ondrejka, <a href="http://www.utm.edu/research/primes/lists/top_ten/">The Top Ten: a Catalogue of Primal Configurations</a>

%H Titus Piezas III, 2004. <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=;rep=rep1&amp;type=pdf">Solving Solvable Sextics Using Polynomial Decomposition</a>

%H Maxie D. Schmidt, <a href="https://arxiv.org/abs/1701.04741">New Congruences and Finite Difference Equations for Generalized Factorial Functions</a>, arXiv:1701.04741 [math.CO], 2017.

%H Apoloniusz Tyszka, <a href="https://hal.archives-ouvertes.fr/hal-01625653/document">A conjecture which implies that there are infinitely many primes of the form n!+1</a>, Preprint, 2017.

%H Apoloniusz Tyszka, <a href="https://hal.archives-ouvertes.fr/hal-01614087v5/document">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</a>, 2018.

%H Apoloniusz Tyszka, <a href="https://philarchive.org/rec/TYSDAS">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)</a>, 2019.

%H Apoloniusz Tyszka, <a href="https://doi.org/10.13140/RG.2.2.28996.68486">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</a>, (2019).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FactorialPrime.html">Factorial Prime</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

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

%t v = {0, 1, 2}; Do[If[ !PrimeQ[n + 1] && PrimeQ[n! + 1], v = Append[v, n]; Print[v]], {n, 3, 29651}]

%t Select[Range[100], PrimeQ[#! + 1] &] (* _Alonso del Arte_, Jul 24 2014 *)

%o (PARI) for(n=0,500,if(ispseudoprime(n!+1),print1(n", "))) \\ _Charles R Greathouse IV_, Jun 16 2011

%o (MAGMA) [n: n in [0..800] | IsPrime(Factorial(n)+1)]; // _Vincenzo Librandi_, Oct 31 2018

%o (Python)

%o from sympy import factorial, isprime

%o for n in range(0,800):

%o if isprime(factorial(n)+1):

%o print(n, end=', ') # _Stefano Spezia_, Jan 10 2019

%Y Cf. A002982 (n!-1 is prime), A064295. A088332 gives the primes.

%Y Equals A090660 - 1.

%Y Cf. A093804.

%K hard,more,nonn,nice

%O 1,3

%A _N. J. A. Sloane_

%E Term 6380 sent in by _Jud McCranie_, May 08 2000

%E Term 26951 from Ken Davis (kraden(AT)ozemail.com.au), May 24 2002

%E Term 110059 found by PrimeGrid around Jun 11 2011, submitted by _Eric W. Weisstein_, Jun 13 2011

%E Term 150209 by _Rene Dohmen_, Jun 09 2012

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Last modified April 15 07:23 EDT 2021. Contains 342975 sequences. (Running on oeis4.)