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A002982 Numbers n such that n! - 1 is prime.
(Formerly M2321)
86

%I M2321

%S 3,4,6,7,12,14,30,32,33,38,94,166,324,379,469,546,974,1963,3507,3610,

%T 6917,21480,34790,94550,103040,147855,208003

%N Numbers n such that n! - 1 is prime.

%C The corresponding primes n!-1 are often called factorial primes.

%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 166, p. 53, Ellipses, Paris 2008.

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

%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., 26:118 (1972), pp. 567-570.

%H J. P. Buhler et al., <a href="http://dx.doi.org/10.1090/S0025-5718-1982-0645679-1">Primes of the form n!+-1 and 2.3.5....p+-1</a>, Math. Comp., 38:158 (1982), pp. 639-643.

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

%H C. 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., 71:237 (2002), pp. 441-448.

%H P. Carmody, <a href="http://83.143.57.194:16384/Factorial/">Factorial Prime Search Progress Pages</a>

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

%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 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 (2015), pp 213-219. doi:10.1017/mag.2015.28.

%H R. 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.

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

%H PrimeGrid, <a href="http://www.primegrid.com/forum_thread.php?id=1336&amp;nowrap=true#26809">World Record Factorial Prime!!!</a>

%H PrimeGrid, <a href="http://www.primegrid.com/download/fps-94550.pdf">Announcement of 94550</a>, (2010) - _Felix Fröhlich_, Jul 11 2014

%H PrimeGrid, <a href="http://www.primegrid.com/download/fps-103040.pdf">Announcement of 103040</a>, (2010) - _Felix Fröhlich_, Jul 11 2014

%H PrimeGrid, <a href="http://www.primegrid.com/download/FPS-147855.pdf">Announcement of 147855</a>, (2013) - _Felix Fröhlich_, Jul 11 2014

%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/Factorial.html">Factorial</a>

%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 R. G. Wilson, V, <a href="/A002982/a002982.pdf">Letter to N. J. A. Sloane, Jan. 1989</a>

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

%e From _Gus Wiseman_, Jul 04 2019: (Start)

%e The sequence of numbers n! - 1 together with their prime indices begins:

%e 1: {}

%e 5: {3}

%e 23: {9}

%e 119: {4,7}

%e 719: {128}

%e 5039: {675}

%e 40319: {9,273}

%e 362879: {5,5,430}

%e 3628799: {10,11746}

%e 39916799: {6,7,9,992}

%e 479001599: {25306287}

%e 6227020799: {270,256263}

%e 87178291199: {3610490805}

%e 1307674367999: {7,11,11,16,114905}

%e 20922789887999: {436,318519035}

%e 355687428095999: {8,21,10165484947}

%e 6402373705727999: {17,20157,25293727}

%e 121645100408831999: {119,175195,4567455}

%e 2432902008176639999: {11715,659539127675}

%e (End)

%t Select[Range[10^3], PrimeQ[ #!-1] &] (* _Vladimir Joseph Stephan Orlovsky_, May 01 2008 *)

%o (PARI) is(n)=ispseudoprime(n!-1) \\ _Charles R Greathouse IV_, Mar 21 2013

%o (MAGMA) [n: n in [0..500] | IsPrime(Factorial(n)-1)]; // _Vincenzo Librandi_, Sep 07 2017

%o (Python)

%o from sympy import factorial, isprime

%o A002982_list = [n for n in range(1,10**2) if isprime(factorial(n)-1)] # _Chai Wah Wu_, Apr 04 2021

%Y Cf. A002981 (numbers n such that n!+1 is prime).

%Y Cf. A055490 (primes of form n!-1).

%Y Cf. A088332 (primes of form n!+1).

%Y Cf. A000142, A001221, A001222, A046051, A054991, A112798, A325272.

%K hard,more,nonn,nice

%O 1,1

%A _N. J. A. Sloane_

%E 21480 sent in by Ken Davis (ken.davis(AT)softwareag.com), Oct 29 2001

%E Updated Feb 26 2007 by _Max Alekseyev_, based on progress reported in the Carmody web site.

%E Inserted missing 21480 and 34790 (see Caldwell). Added 94550, discovered Oct 05 2010. _Eric W. Weisstein_, Oct 06 2010

%E 103040 was discovered by James Winskill, Dec 14 2010. It has 471794 digits. Corrected by _Jens Kruse Andersen_, Mar 22 2011

%E a(26) = 147855 from _Felix Fröhlich_, Sep 02 2013

%E a(27) = 208003 from _Sou Fukui_, Jul 27 2016

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Last modified April 19 09:12 EDT 2021. Contains 343110 sequences. (Running on oeis4.)