login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A002981 Numbers n such that n! + 1 is prime.
(Formerly M0908)
93

%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 n such that n! + 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=10.1.1.90.6646&amp;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 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=0,1e4,if(ispseudoprime(n!+1),print1(n", "))) \\ _Charles R Greathouse IV_, Jun 16 2011

%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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 18 14:30 EST 2017. Contains 296177 sequences.