A055490 -id:A055490 - cp's OEIS frontend

A002982 Numbers k such that k! - 1 is prime.

3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, 103040, 147855, 208003

Offset: 1

N. J. A. Sloane

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

			From _Gus Wiseman_, Jul 04 2019: (Start)
The sequence of numbers n! - 1 together with their prime indices begins:
                    1: {}
                    5: {3}
                   23: {9}
                  119: {4,7}
                  719: {128}
                 5039: {675}
                40319: {9,273}
               362879: {5,5,430}
              3628799: {10,11746}
             39916799: {6,7,9,992}
            479001599: {25306287}
           6227020799: {270,256263}
          87178291199: {3610490805}
        1307674367999: {7,11,11,16,114905}
       20922789887999: {436,318519035}
      355687428095999: {8,21,10165484947}
     6402373705727999: {17,20157,25293727}
   121645100408831999: {119,175195,4567455}
  2432902008176639999: {11715,659539127675}
(End)

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 166, p. 53, Ellipses, Paris 2008.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 118.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 719 at p. 160.

A. Borning, Some results for k!+-1 and 2.3.5...p+-1, Math. Comp., 26:118 (1972), pp. 567-570.
J. P. Buhler et al., Primes of the form n!+-1 and 2.3.5....p+-1, Math. Comp., 38:158 (1982), pp. 639-643.
Chris K. Caldwell, Factorial Primes.
C. K. Caldwell and Y. Gallot, On the primality of n!+-1 and 2*3*5*...*p+-1, Math. Comp., 71:237 (2002), pp. 441-448.
P. Carmody, Factorial Prime Search Progress Pages.
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.
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
R. K. Guy and N. J. A. Sloane, Correspondence, 1985.
H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3, 1987), 197-203. (Annotated scanned copy)
Des MacHale and Joseph Manning, Maximal runs of strictly composite integers, The Mathematical Gazette, 99 (2015), pp 213-219. doi:10.1017/mag.2015.28.
R. 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.
R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations.
PrimeGrid, World Record Factorial Prime!!!.
PrimeGrid, Announcement of 94550, (2010). - _Felix Fröhlich_, Jul 11 2014
PrimeGrid, Announcement of 103040, (2010). - _Felix Fröhlich_, Jul 11 2014
PrimeGrid, Announcement of 147855, (2013). - _Felix Fröhlich_, Jul 11 2014
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Eric Weisstein's World of Mathematics, Factorial.
Eric Weisstein's World of Mathematics, Factorial Prime.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jan. 1989.
Index entries for sequences related to factorial numbers.

Cf. A002981 (numbers n such that n!+1 is prime).
Cf. A055490 (primes of form n!-1).
Cf. A088332 (primes of form n!+1).
Cf. A000142, A001221, A001222, A046051, A054991, A112798, A325272.

[n: n in [0..500] | IsPrime(Factorial(n)-1)]; // Vincenzo Librandi, Sep 07 2017

Select[Range[10^3], PrimeQ[ #!-1] &] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)

is(n)=ispseudoprime(n!-1) \\ Charles R Greathouse IV, Mar 21 2013

from sympy import factorial, isprime
A002982_list = [n for n in range(1,10**2) if isprime(factorial(n)-1)] # Chai Wah Wu, Apr 04 2021

21480 sent in by Ken Davis (ken.davis(AT)softwareag.com), Oct 29 2001
Updated Feb 26 2007 by Max Alekseyev, based on progress reported in the Carmody web site.
Inserted missing 21480 and 34790 (see Caldwell). Added 94550, discovered Oct 05 2010. Eric W. Weisstein, Oct 06 2010
103040 was discovered by James Winskill, Dec 14 2010. It has 471794 digits. Corrected by Jens Kruse Andersen, Mar 22 2011
a(26) = 147855 from Felix Fröhlich, Sep 02 2013
a(27) = 208003 from Sou Fukui, Jul 27 2016

A104364 Primes of the form A104350(k) - 1.

Original entry on oeis.org

5, 11, 59, 179, 1259, 7559, 37799, 415799, 1135133999, 5499724229999, 29220034833989999, 1408101540804746673385499999, 43673268652925265723884051023987499999

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..26

Intersection of A104357 and A000040.

Cf. A104372, A104373, A055490, A057705.

Select[FoldList[Times, Array[FactorInteger[#][[-1, 1]] &, 100]] - 1, PrimeQ] (* Amiram Eldar, Apr 08 2024 *)

gpf(n) = {my(p = factor(n)[, 1]); if(n == 1, 1, p[#p]);}
lista(nmax) = {my(r = 1); for(k = 1, nmax, r * = gpf(k); if(isprime(r-1), print1(r-1, ", ")));} \\ Amiram Eldar, Apr 08 2024

A088054 Factorial primes: primes which are within 1 of a factorial number.

Original entry on oeis.org

2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999

Offset: 1

Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Nov 02 2003

Conjecture: 3 is the intersection of A002981 and A002982.

			3! + 1 = 7; 7! - 1 = 5039.
39916801 is a term because 11! + 1 is prime.

Chai Wah Wu, Table of n, a(n) for n = 1..29
C. Caldwell's The Top Twenty, Factorial Primes.
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
Romeo Meštrović, 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], 2012-2018. - From _N. J. A. Sloane_, Jun 13 2012
Wikipedia, Factorial prime.

Cf. A000142, A002981, A002982.

Union of A055490 and A088332.

t = {}; Do[ If[PrimeQ[n! - 1], AppendTo[t, n! - 1]]; If[PrimeQ[n! + 1], AppendTo[t, n! + 1]], {n, 50}]; t (* Robert G. Wilson v *)
Union[Select[Range[50]!-1, PrimeQ], Select[Range[50]!+1, PrimeQ]] (Noe)
fp[n_] := Module[{nf=n!}, Select[{nf-1,nf+1},PrimeQ]]; Flatten[ Table[ fp[i],{i,50}]] (* Harvey P. Dale, Dec 18 2010 *)
Select[Flatten[#+{-1,1}&/@(Range[50]!)],PrimeQ] (* Harvey P. Dale, Apr 08 2019 *)

from itertools import count, islice
from sympy import isprime
def A088054_gen(): # generator of terms
    f = 1
    for k in count(1):
        f *= k
        if isprime(f-1):
            yield f-1
        if isprime(f+1):
            yield f+1
A088054_list = list(islice(A088054_gen(),10)) # Chai Wah Wu, Feb 18 2022

Corrected by Paul Muljadi, Oct 11 2005

More terms from Robert G. Wilson v and T. D. Noe, Oct 12 2005

A163076 Primes of the form k$ - 1. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

5, 19, 29, 139, 251, 12011, 48619, 51479, 155117519, 81676217699, 1378465288199, 5651707681619, 386971244197199, 1580132580471899, 30067266499541039, 6637553085023755473070799, 35257120210449712895193719, 399608854866744452032002440111

Offset: 1

Peter Luschny, Jul 21 2009

nonn

			Since 4$ = 6 the prime 5 is listed.

Jinyuan Wang, Table of n, a(n) for n = 1..53
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A055490, A056040, A163078 (arguments k), A163074, A163075.

a := proc(n) select(isprime, map(x -> A056040(x)-1,[$1..n])); sort(%) end:

Reap[Do[f = n!/Quotient[n, 2]!^2; If[PrimeQ[p = f - 1], Sow[p]], {n, 1, 70}]][[2, 1]] // Union (* Jean-François Alcover, Jun 28 2013 *)

More terms from Jinyuan Wang, Mar 22 2020

A062701 Index of factorial primes of the form k! + 1.

Original entry on oeis.org

1, 2, 4, 2428957

Offset: 1

Labos Elemer, Jul 11 2001

			The exact subscript of the 5th prime [1 + 27! = 10888869450418352160768000001] is not yet available.

Cf. A000720, A002981, A002982, A055490, A088332.

a(n) = PrimePi(A002981(n)!+1).

Offset 1 from Michel Marcus, Aug 29 2019

A062702 Index of factorial primes of form m!-1.

Original entry on oeis.org

3, 9, 128, 675, 25306287, 3610490805

Offset: 1

Labos Elemer, Jul 11 2001

			The exact subscript of 7th prime [=30!-1=265252859812191058636308479999999] is not yet available.

Cf. A002981, A002982, A055490.

a(n) = PrimePi(A002982(n)!-1) = A000720(A055490(n)).

Offset 1 from Michel Marcus, Aug 29 2019

A093622 Largest prime of the form n!/k!-1.

Original entry on oeis.org

5, 23, 59, 719, 5039, 6719, 181439, 5039, 6652799, 479001599, 154439, 87178291199, 54486431999, 3487131647999, 59281238015999, 1067062284287999, 1013709170073599, 405483668029439999, 39070079, 180503769599

Offset: 3

Hugo Pfoertner, Apr 06 2004

nonn

			a(5) =59 because 5!/1!-1=119=7*17 is composite, whereas 5!/2!-1=59 is prime.

Dario Alpern, Factorization using the Elliptic Curve Method.

Cf. A093623 smallest k>0 such that n!/k!-1 is prime, A055490 primes of form n!-1, A093437 largest prime of the form n!/k!+1.

A233011 Primes of the form (2*n)! - n!^2 - 1.

Original entry on oeis.org

19, 683, 478483199, 20921164185599

Offset: 1

K. D. Bajpai, Dec 03 2013

nonn

The 5th term a(5) has 268 digits and is too long to display in data section.
The 7th term a(7) in the sequence has 823 digits.
a(8) has 2030 digits; a(9) has 2264 digits (these are not included in b-file).

			a(1)= 19: n= 2: (2*n)!- n!^2-1= 19 which is prime.
a(2)= 683: n= 3: (2*n)!- n!^2-1= 683 which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..7

Cf. A055490 (primes: n! -1).

Cf. A118812 (primes: (2*n)!-n!+1).

KD := proc() local a; a:=(2*n)!-n!^2-1; if isprime(a) then RETURN (a);  fi; end: seq(KD(), n=1..200);

A374901 Numbers k such that k!^2 + ((k - 1)!^2) + 1 is prime.

Original entry on oeis.org

1, 3, 4, 6, 10, 11, 118, 271, 288, 441, 457, 2931, 5527, 6984, 9998, 10395, 13703

Offset: 1

Arsen Vardanyan, Jul 31 2024

a(18) > 15000 - Karl-Heinz Hofmann, Aug 23 2024

			4 is a term, because 4!^2 + 3!^2 + 1 = 576 + 36 + 1 = 613 is a prime number.

Cf. A000142, A055490, A358805, A358878, A359180, A080778, A243078, A242994

PARI
```
is(k) = isprime((k!^2)+((k-1)!)^2+1);
```

from itertools import count, islice
from sympy import isprime
def A374901_gen(): # generator of terms
    f = 1
    for k in count(1):
        if isprime((k**2+1)*f+1):
            yield k
        f *= k**2
A374901_list = list(islice(A374901_gen(),10)) # Chai Wah Wu, Oct 02 2024

a(12)-a(14) from Michael S. Branicky, Aug 01 2024

a(15)-a(17) from Karl-Heinz Hofmann, Aug 23 2024

A126782 Primes of the form [n! mod (n!!+1)]/2, with n>=1.

Original entry on oeis.org

3, 17, 29, 281, 254993, 690953, 607435538171963, 192133794380608031505991200873083839505054136751452696277424837839455632569607117048950195313

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Mar 14 2007

The next term has 611 digits. - Harvey P. Dale, Apr 15 2018

			n=6 n!=720 n!!=48 [n! mod (n!!+1)]/2 = (720 mod 49)/2 = 34/2 = 17
n=7 n!=5040 n!!=105 [n! mod (n!!+1)]/2 = (5040 mod 106)/2 = 58/2 = 29

Cf. A055490.

P:=proc(n) local i,j,k,w; for i from 2 by 1 to n do k:=i; w:=i-2; while w>0 do k:=k*w; w:=w-2; od; j:=(i! mod (k+1))/2; if isprime(j) then print(j); fi; od; end: P(1000);

Select[Table[Mod[n!,n!!+1]/2,{n,200}],PrimeQ] (* Harvey P. Dale, Apr 15 2018 *)

cp's OEIS Frontend

A002982 Numbers k such that k! - 1 is prime.

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A104364 Primes of the form A104350(k) - 1.

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A088054 Factorial primes: primes which are within 1 of a factorial number.

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A163076 Primes of the form k$ - 1. Here '$' denotes the swinging factorial function (A056040).

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Maple

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A062701 Index of factorial primes of the form k! + 1.

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A062702 Index of factorial primes of form m!-1.

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A093622 Largest prime of the form n!/k!-1.

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A233011 Primes of the form (2*n)! - n!^2 - 1.

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