A088332 -id:A088332 - cp's OEIS frontend

A002981 Numbers k such that k! + 1 is prime.

0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, 288465, 308084, 422429

Offset: 1

N. J. A. Sloane

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

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

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).
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 p. 70.

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, 110059! + 1 on the Prime Pages.
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 and N. J. A. Sloane, Correspondence, 1991.
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.
N. Kuosa, Source for 6380.
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.
Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations.
Titus Piezas III, 2004. Solving Solvable Sextics Using Polynomial Decomposition.
PrimePages, Factorial Primes.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Apoloniusz Tyszka, A conjecture which implies that there are infinitely many primes of the form n!+1, Preprint, 2017.
Apoloniusz Tyszka, 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, 2018.
Apoloniusz Tyszka, 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), 2019.
Apoloniusz Tyszka, 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, (2019).
Eric Weisstein's World of Mathematics, Factorial Prime.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.
Index entries for sequences related to factorial numbers.

Cf. A002982 (n!-1 is prime), A064295. A088332 gives the primes.
Equals A090660 - 1.
Cf. A093804.

[n: n in [0..800] | IsPrime(Factorial(n)+1)]; // Vincenzo Librandi, Oct 31 2018

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 *)

for(n=0,500,if(ispseudoprime(n!+1),print1(n", "))) \\ Charles R Greathouse IV, Jun 16 2011

from sympy import factorial, isprime
for n in range(0,800):
    if isprime(factorial(n)+1):
        print(n, end=', ') # Stefano Spezia, Jan 10 2019

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
a(24)-a(25) from Dmitry Kamenetsky, Jun 19 2024

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

Original entry on oeis.org

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

A117141 Primes of the form n!! - 1.

Original entry on oeis.org

2, 7, 47, 383, 10321919, 51011754393599, 1130138339199322632554990773529330319359999999, 73562883979319395645666688474019139929848516028923903999999999, 4208832729023498248022825567687608993477547383960134557368319999999999

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Apr 21 2006

nonn

			6!! - 1 = 6*4*2 - 1 = 48 - 1 = 47, which is prime.
8!! - 1 = 8*6*4*2 - 1 = 384 - 1 = 383, which is prime.

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 158.

Vincenzo Librandi, Table of n, a(n) for n = 1..15

Cf. A002981, A002982, A006882, A007749, A088332.

Cf. A093173 = primes of the form (2^n * n!) - 1.

SFACT:= proc(n) local i,j,k; for k from 1 by 1 to n do i:=k; j:=k-2; while j >0 do i:=i*j; j:=j-2; od: if isprime(i-1) then print(i-1); fi; od: end: SFACT(100);

lst={};Do[p=n!!-1;If[PrimeQ[p],AppendTo[lst,p]],{n,0,5!,1}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)
Select[Table[n!!-1,{n,1,100}],PrimeQ] (* Vincenzo Librandi, Dec 07 2011 *)

print1(2);for(n=1, 1e3, if(ispseudoprime(t=n!<Charles R Greathouse IV, Jun 16 2011

a(n) = A093173(n-1) for n > 1. - Alexander Adamchuk, Apr 18 2007

a(n) = A006882(A007749(n)) - 1. - Elmo R. Oliveira, Feb 22 2025

A104372 Primes of the form A104350(k) + 1.

Original entry on oeis.org

2, 3, 7, 13, 61, 181, 2521, 7561, 415801, 1247401, 1099944846001, 146100174169950001, 1156675078903494150001, 750321420485151941966263672363958662088980270355720625000001

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..16

Intersection of A104365 and A000040.

Cf. A104364, A104373, A088332, A018239.

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

a(14) from Amiram Eldar, Apr 09 2024

A125162 a(n) is the number of primes of the form k! + n, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 4, 0, 1, 1, 5, 1, 3, 0, 1, 1, 6, 1, 7, 0, 1, 1, 6, 0, 1, 0, 1, 1, 6, 1, 9, 0, 0, 0, 3, 1, 11, 0, 1, 1, 9, 1, 5, 0, 1, 1, 10, 0, 2, 0, 1, 1, 9, 0, 2, 0, 1, 1, 10, 1, 9, 0, 0, 0, 3, 1, 9, 0, 1, 1, 8, 1, 9, 0, 0, 0, 5, 1, 9, 0, 1, 1, 11, 0, 1, 0, 1, 1, 8, 0, 3, 0, 0, 0, 2, 1, 10, 0, 1, 1, 10, 1

Offset: 1

Alexander Adamchuk, Nov 21 2006

nonn

Note the triples of consecutive zeros in a(n) for n = {{32,33,34}, {62,63,64}, {74,75,76}, {92,93,94}, {116,117,118}, {122,123,124}, {140,141,142}, {152,153,154}, {158,159,160}, {182,183,184}, {200,201,202}, {206,207,208}, {212,213,214}, {218,219,220}, {242,243,244}, {272,273,274}, {284,285,286}, ...}. The middle index of most zero triples is a multiple of 3. See A125164.
The first consecutive quintuple of zeros has indices n = {294,295,296,297,298}, where the odd zero index n = 295 is not a multiple of 3.
Also for n >= 2, a(n) is the number of primes of the form k! + n for all k, since n divides k! + n for k >= n. Note that it is not known whether there are infinitely many primes of the form k! + 1; see A088332 for such primes and A002981 for the indices k. - Jianing Song, Jul 28 2018

			a(n) is the length of n-th row in the table of numbers k such that k! + n is a prime, 1 <= k <= n.
   n:  numbers k
   -------------
   1:  {1},
   2:  {1},
   3:  {2},
   4:  {1},
   5:  {2, 3, 4},
Thus a(1)-a(4) = 1, a(5) = 3.
See Example table link for more rows.

Michel Marcus, Example table

Cf. A125163 (indices of 0), A125164 (triples).

Table[Length[Select[Range[n],PrimeQ[ #!+n]&]],{n,1,300}]

a(n)=c=0;for(k=1,n,if(ispseudoprime(k!+n),c++));c
vector(100,n,a(n)) \\ Derek Orr, Oct 15 2014

Name clarified by Jianing Song, Jul 28 2018

Edited by Michel Marcus, Jul 29 2018

A136437 a(n) = prime(n) - k! where k is the greatest number such that k! <= prime(n).

Original entry on oeis.org

0, 1, 3, 1, 5, 7, 11, 13, 17, 5, 7, 13, 17, 19, 23, 29, 35, 37, 43, 47, 49, 55, 59, 65, 73, 77, 79, 83, 85, 89, 7, 11, 17, 19, 29, 31, 37, 43, 47, 53, 59, 61, 71, 73, 77, 79, 91, 103, 107, 109, 113, 119, 121, 131, 137, 143, 149, 151, 157, 161, 163, 173, 187, 191, 193, 197, 211, 217, 227, 229, 233, 239, 247

Offset: 1

Ctibor O. Zizka, Apr 02 2008

How many times does each prime appear in this sequence?
The only value (prime(n) - k!) = 0 is at n=1, where k=2.
Are n=2, k=2 and n=4, k=3 the only occurrences of (prime(n) - k!) = 1?
There exist infinitely many solutions of the form (prime(n) - k!) = prime(n-t), t < n.
Are there infinitely many solutions of the form (prime(n) - k!) = prime(r_1)*...*prime(r_i); r_i < n for all i?
From Bernard Schott, Jul 16 2021: (Start)
Answer to the second question is no: 18 other occurrences (n,k) of (prime(n) - k!) = 1 are known today; indeed, every k > 1 in A002981 that satisfies k! + 1 is prime gives an occurrence, but only a third pair (n, k) is known exactly; and this comes for n = 2428957, k = 11 because (prime(2428957) - 11!) = 1.
The next occurrence corresponds to k = 27 and n = X where prime(X) = 1+27! = 10888869450418352160768000001 but index X is not yet available (see A062701).
For the occurrences of (prime(m) - k!) = 1, integers k are in A002981 \ {0, 1}, corresponding indices m are in A062701 \ {1} (only 3 indices are known today) and prime(m) are in A088332 \ {2}. (End)

			a(1)  = prime(1)  - 2! =  2 -  2 =  0;
a(2)  = prime(2)  - 2! =  3 -  2 =  1;
a(3)  = prime(3)  - 2! =  5 -  2 =  3;
a(4)  = prime(4)  - 3! =  7 -  6 =  1;
a(5)  = prime(5)  - 3! = 11 -  6 =  5;
a(6)  = prime(6)  - 3! = 13 -  6 =  7;
a(7)  = prime(7)  - 3! = 17 -  6 = 11;
a(8)  = prime(8)  - 3! = 19 -  6 = 13;
a(9)  = prime(9)  - 3! = 23 -  6 = 17;
a(10) = prime(10) - 4! = 29 - 24 =  5.

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A135996, A000040, A000142, A212598, A212266.

Cf. also A002981, A062701, A088332, A346425 (gives k).

f:=proc(n) local p,i; p:=ithprime(n); for i from 0 to p do if i! > p then break; fi; od; p-(i-1)!; end;
[seq(f(n),n=1..70)]; # N. J. A. Sloane, May 22 2012

a[n_] := Module[{p, k},p = Prime[n];k = 1;While[Factorial[k] <= p, k++];p - Factorial[k - 1]] (* James C. McMahon, May 05 2025 *)

a(n) = my(k=1, p=prime(n)); while (k! <= p, k++); p - (k-1)!; \\ Michel Marcus, Feb 19 2019

a(n) = prime(n)- k! where k is the greatest number for which k! <= prime(n).
a(n) = A212598(prime(n)). - Michel Marcus, Feb 19 2019
a(n) = A000040(n) - A346425(n). - Bernard Schott, Jul 16 2021

More terms from Jinyuan Wang, Feb 18 2019

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

A089359 Primes which can be partitioned into distinct factorials. 0! and 1! are not considered distinct.

Original entry on oeis.org

2, 3, 7, 31, 127, 151, 727, 751, 5167, 5791, 5881, 40351, 40471, 41047, 41161, 45361, 45481, 362911, 363751, 368047, 368647, 368791, 403327, 403951, 408241, 408271, 408361, 409081, 3628927, 3629671, 3633991, 3634591, 3669241, 3669847, 3669961

Offset: 1

Amarnath Murthy, Nov 07 2003

nonn

			From _Seiichi Manyama_, Mar 24 2018: (Start)
n | a(n) |
--+------+------------------
1 |    2 | 2!
2 |    3 | 2! + 1!
3 |    7 | 3! + 1!
4 |   31 | 4! + 3! + 1!
5 |  127 | 5! + 3! + 1!
6 |  151 | 5! + 4! + 3! + 1! (End)

Alois P. Heinz, Table of n, a(n) for n = 1..16812 (first 1000 terms from Seiichi Manyama)

Cf. A059590, A088332, A300947, A301593.

from sympy import isprime
def facbase(k, f):
    return sum(f[i] for i, bi in enumerate(bin(k)[2:][::-1]) if bi == "1")
def auptoN(N): # terms up to N factorial-base digits; 20 generates b-file
    f = [factorial(i) for i in range(1, N+1)]
    return list(filter(isprime, (facbase(k, f) for k in range(2**N))))
print(auptoN(10)) # Michael S. Branicky, Oct 15 2022

More terms from Vladeta Jovovic, Nov 08 2003

A093804 Primes p such that p! + 1 is also prime.

Original entry on oeis.org

2, 3, 11, 37, 41, 73, 26951, 110059, 150209

Offset: 1

Jason Earls, May 19 2004

Or, numbers n such that Sum_{d|n} d! is prime.
The prime 26951 from A002981 (n!+1 is prime) is a member since Sum_{d|n} d! = n!+1 if n is prime. - Jonathan Sondow, Jan 30 2005
a(n) are the primes in A002981[n] = {0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, ...} Numbers n such that n! + 1 is prime. Corresponding primes of the form p! + 1 are listed in A103319[n] = {3, 7, 39916801, 13763753091226345046315979581580902400000001, 33452526613163807108170062053440751665152000000001, ...}. - Alexander Adamchuk, Sep 23 2006

			Sum_{d|3} d! = 1! + 3! = 7 is prime, so 3 is a member.

Chris K. Caldwell, The List of Largest Known Primes, 110059! + 1
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv:1202.3670 [math.HO], 2012. - From N. J. A. Sloane, Jun 13 2012
Apoloniusz Tyszka, 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, 2018.
Apoloniusz Tyszka, A new approach to solving number theoretic problems, 2018.

Cf. A062363, A002981, A038507, A088332, A103317, A103319.

seq(`if`(isprime(ithprime(n)!+1), ithprime(n), NULL),n=1..25); # Nathaniel Johnston, Jun 28 2011

Select[Prime[Range[5! ]],PrimeQ[ #!+1]&] (* Vladimir Joseph Stephan Orlovsky, Nov 17 2009 *)

isok(n) = ispseudoprime(n) && ispseudoprime(n!+1); \\ Jinyuan Wang, Jan 20 2020

One more term from Alexander Adamchuk, Sep 23 2006
a(8)=110059 (found on Jun 11 2011, by PrimeGrid), added by Arkadiusz Wesolowski, Jun 28 2011
a(9)=150209 (found on Jun 09 2012, by Rene Dohmen), added by Jinyuan Wang, Jan 20 2020

A163075 Primes of the form k$ + 1. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

2, 3, 7, 31, 71, 631, 3433, 51481, 2704157, 280816201, 4808643121, 35345263801, 2104098963721, 94684453367401, 1580132580471901, 483701705079089804581, 6892620648693261354601, 410795449442059149332177041, 2522283613639104833370312431401

Offset: 1

Peter Luschny, Jul 21 2009

nonn

			Since 3$ = 4$ = 6 the prime 7 is listed, however only once.

Jinyuan Wang, Table of n, a(n) for n = 1..50
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A056040, A088332, A163077 (arguments k), A163074, A163076.

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

cp's OEIS Frontend

A002981 Numbers k such that k! + 1 is prime.

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

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A117141 Primes of the form n!! - 1.

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A104372 Primes of the form A104350(k) + 1.

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A125162 a(n) is the number of primes of the form k! + n, 1 <= k <= n.

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A136437 a(n) = prime(n) - k! where k is the greatest number such that k! <= prime(n).

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