A002981 -id:A002981 - cp's OEIS frontend

A038507 a(n) = n! + 1.

2, 2, 3, 7, 25, 121, 721, 5041, 40321, 362881, 3628801, 39916801, 479001601, 6227020801, 87178291201, 1307674368001, 20922789888001, 355687428096001, 6402373705728001, 121645100408832001, 2432902008176640001, 51090942171709440001, 1124000727777607680001, 25852016738884976640001

Offset: 0

N. J. A. Sloane

"For n = 4, 5 and 7, n!+1 is a square. Sierpiński asked if there are any other values of n with this property." p. 82 of Ogilvy and Anderson (see A146968).
Number of {12,12*,1*2,21*,2*1}-avoiding signed permutations in the hyperoctahedral group.
After Wilson's Theorem: if (n+1) is prime then (n+1) is the smallest prime factor of a(n). - Karl-Heinz Hofmann, Aug 21 2024

			G.f. = 2 + 2*x + 3*x^2 + 7*x^3 + 25*x^4 + 121*x^5 + 721*x^6 + 5041*x^7 + ...

C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, 1966, p. 82.
Wacław Sierpiński, On some unsolved problems of arithmetics, Scripta Mathematica, vol. 25 (1960), p. 125.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 763 and Encyclopedia of Combinatorial Structures 834
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
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], 2012-2023. - From _N. J. A. Sloane_, Jun 13 2012
Gerard P. Michon, Wilson's Theorem
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Andrew Walker, Factors of n! +- 1
Arthur T. White, Ringing the changes, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 2, 203-215.
Robert G. Wilson v, Explicit factorizations
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 4.
Index entries for sequences related to factorial numbers

Cf. A000142, A002583, A002981, A033312, A051301, A056111, A146968, A217702.

a038507 = (+ 1) . a000142
a038507_list = 2 : f 1 2 where
   f x y = z : f (x + 1) z where z = x * (y - 1) + 1
-- Reinhard Zumkeller, Mar 20 2013

[Factorial(n) +1: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011

Range[0,20]!+1 (* Harvey P. Dale, May 06 2012 *)

A038507(n):= n!+1$
makelist(A038507(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

a(n)=n!+1 \\ Charles R Greathouse IV, Nov 20 2012

from math import factorial
def A038507(n): return factorial(n) + 1 # Karl-Heinz Hofmann, Aug 21 2024

[factorial(n) + 1 for n in range(0,24)] # Stefano Spezia, Apr 21 2025

a(n) = n * (a(n-1) - 1) + 1. - Reinhard Zumkeller, Mar 20 2013
0 = a(n)*(a(n+1) - 5*a(n+2) + 5*a(n+3) - a(n+4)) + a(n+1)*(a(n+1) + a(n+2) - 6*a(n+3) + 2*a(n+4)) + a(n+2)*(3*a(n+2) - a(n+3) - a(n+4)) + a(n+3)*(a(n+3)) if n>=0. - Michael Somos, Apr 23 2014
From Ilya Gutkovskiy, Jan 20 2017: (Start)
E.g.f: exp(x) + 1/(1 - x).
Sum_{n>=0} 1/a(n) = A217702. (End)

Additional comments from Jason Earls, Apr 01 2001
Numericana.com URL fixed by Gerard P. Michon, Mar 30 2010
Entry revised by N. J. A. Sloane, Jun 10 2012

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

A002109 Hyperfactorials: Product_{k = 1..n} k^k.

Original entry on oeis.org

1, 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, 55696437941726556979200000, 21577941222941856209168026828800000, 215779412229418562091680268288000000000000000, 61564384586635053951550731889313964883968000000000000000

Offset: 0

N. J. A. Sloane

A054374 gives the discriminants of the Hermite polynomials in the conventional (physicists') normalization, and A002109 (this sequence) gives the discriminants of the Hermite polynomials in the (in my opinion more natural) probabilists' normalization. See refs Wikipedia and Szego, eq. (6.71.7). - Alan Sokal, Mar 02 2012
a(n) = (-1)^n/det(M_n) where M_n is the n X n matrix m(i,j) = (-1)^i/i^j. - Benoit Cloitre, May 28 2002
a(n) = determinant of the n X n matrix M(n) where m(i,j) = B(n,i,j) and B(n,i,x) denote the Bernstein polynomial: B(n,i,x) = binomial(n,i)*(1-x)^(n-i)*x^i. - Benoit Cloitre, Feb 02 2003
Partial products of A000312. - Reinhard Zumkeller, Jul 07 2012
Number of trailing zeros (A246839) increases every 5 terms since the exponent of the factor 5 increases every 5 terms and the exponent of the factor 2 increases every 2 terms. - Chai Wah Wu, Sep 03 2014
Also the number of minimum distinguishing labelings in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017
Also shows up in a term in the solution to the generalized version of Raabe's integral. - Jibran Iqbal Shah, Apr 24 2021

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 135-145.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 477.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
G. Szego, Orthogonal Polynomials, American Mathematical Society, 1981 edition, 432 Pages.

N. J. A. Sloane, Table of n, a(n) for n = 0..36
Christian Aebi and Grant Cairns, Generalizations of Wilson's Theorem for Double-, Hyper-, Sub-and Superfactorials, The American Mathematical Monthly 122.5 (2015): 433-443.
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537. Zbl 1133.11012.
blackpenredpen, What is a Hyperfactorial? Youtube video (2018).
CreativeMathProblems, A beautiful integral | Raabe's integral, Youtube Video (2021).
Steven R. Finch, Glaisher-Kinkelin Constant (gives asymptotic expressions for A002109, A000178) [Broken link]
Steven R. Finch, Glaisher-Kinkelin Constant (gives asymptotic expressions for A002109, A000178) [From the Wayback machine]
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
A. M. Ibrahim, Extension of factorial concept to negative numbers, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.
Jean-Christophe Pain, Series representations for the logarithm of the Glaisher-Kinkelin constant, arXiv:2304.07629 [math.NT], 2023.
Jean-Christophe Pain, Bounds on the p-adic valuation of the factorial, hyperfactorial and superfactorial, arXiv:2408.00353 [math.NT], 2024. See p. 5.
Vignesh Raman, The Generalized Superfactorial, Hyperfactorial and Primorial functions, arXiv:2012.00882 [math.NT], 2020.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., 332 (2007), 292-314; see Section 5.
László Tóth, Weighted gcd-sum functions, J. Integer Sequences, 14 (2011), Article 11.7.7.
Eric Weisstein's World of Mathematics, Hyperfactorial.
Eric Weisstein's World of Mathematics, K-Function.
Wikipedia, Hermite polynomials.
Index entries for sequences related to factorial numbers.
Index to divisibility sequences.

Cf. A000178, A000142, A000312, A001358, A002981, A002982, A100015, A005234, A014545, A018239, A006794, A057704, A057705, A054374.
Cf. A074962 [Glaisher-Kinkelin constant, also gives an asymptotic approximation for the hyperfactorials].
Cf. A001923, A051675, A240993, A255321, A255323, A255344.
Cf. A246839 (trailing 0's).
Cf. A347345, A347352.
Cf. A261175 (number of digits).

a002109 n = a002109_list !! n
a002109_list = scanl1 (*) a000312_list  -- Reinhard Zumkeller, Jul 07 2012

f := proc(n) local k; mul(k^k,k=1..n); end;
A002109 := n -> exp(Zeta(1,-1,n+1)-Zeta(1,-1));
seq(simplify(A002109(n)),n=0..11); # Peter Luschny, Jun 23 2012

Table[Hyperfactorial[n], {n, 0, 11}] (* Zerinvary Lajos, Jul 10 2009 *)
Hyperfactorial[Range[0, 11]] (* Eric W. Weisstein, Jul 14 2017 *)
Join[{1},FoldList[Times,#^#&/@Range[15]]] (* Harvey P. Dale, Nov 02 2023 *)

a(n)=prod(k=2,n,k^k) \\ Charles R Greathouse IV, Jan 12 2012

a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/prod(j=1,k+1,(1+j^j*x+x*O(x^n)) )), n) \\ Paul D. Hanna, Oct 02 2013

A002109 = [1]
for n in range(1, 10):
    A002109.append(A002109[-1]*n**n) # Chai Wah Wu, Sep 03 2014

a = lambda n: prod(falling_factorial(n,k) for k in (1..n))
[a(n) for n in (0..10)]  # Peter Luschny, Nov 29 2015

a(n)*A000178(n-1) = (n!)^n = A036740(n) for n >= 1.
Determinant of n X n matrix m(i, j) = binomial(i*j, i). - Benoit Cloitre, Aug 27 2003
a(n) = exp(zeta'(-1, n + 1) - zeta'(-1)) where zeta(s, z) is the Hurwitz zeta function. - Peter Luschny, Jun 23 2012
G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1 + k^k*x). - Paul D. Hanna, Oct 02 2013
a(n) = A240993(n) / A000142(n+1). - Reinhard Zumkeller, Aug 31 2014
a(n) ~ A * n^(n*(n+1)/2 + 1/12) / exp(n^2/4), where A = 1.2824271291006226368753425... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Feb 20 2015
a(n) = Product_{k=1..n} ff(n,k) where ff denotes the falling factorial. - Peter Luschny, Nov 29 2015
log a(n) = (1/2) n^2 log n - (1/4) n^2 + (1/2) n log n + (1/12) log n + log(A) + o(1), where log(A) = A225746 is the logarithm of Glaisher's constant. - Charles R Greathouse IV, Mar 27 2020
From Amiram Eldar, Apr 30 2023: (Start)
Sum_{n>=1} 1/a(n) = A347345.
Sum_{n>=1} (-1)^(n+1)/a(n) = A347352. (End)
From Andrea Pinos, Apr 04 2024: (Start)
a(n) = e^(Integral_{x=1..n+1} (x - 1/2 - log(sqrt(2*Pi)) + (n+1-x)*Psi(x)) dx), where Psi(x) is the digamma function.
a(n) = e^(Integral_{x=1..n} (x + 1/2 - log(sqrt(2*Pi)) + log(Gamma(x+1))) dx). (End)

A089085 Numbers k such that (k! + 3)/3 is prime.

Original entry on oeis.org

3, 5, 6, 8, 11, 17, 23, 36, 77, 93, 94, 109, 304, 497, 1330, 1996, 3027, 3053, 4529, 5841, 20556, 26558, 28167

Offset: 1

Cino Hilliard, Dec 05 2003

nonn

a(21) > 20000. The PFGW program has been used to certify all the terms up to a(20), using the "N-1" deterministic test. - Giovanni Resta, Mar 31 2014

Cf. A089131.

Cf. n!/m-1 is a prime: A002982, A082671, A139056, A139199-A139205; n!/m+1 is a prime: A002981, A082672, A089085, A139061, A139058, A139063, A139065, A151913, A137390, A139071 (1<=m<=10).

[n: n in [0..500] | IsPrime((Factorial(n)+3) div 3)]; // Vincenzo Librandi, Dec 12 2011

Select[Range[0, 1400], PrimeQ[(#!+3)/3] &] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

is(n)=ispseudoprime(n!\3+1) \\ Charles R Greathouse IV, Mar 21 2013

More terms from Don Reble, Dec 06 2003
1330 from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008
Typo in Mma program corrected by Vincenzo Librandi, Dec 12 2011
a(16)-a(20) from Giovanni Resta, Mar 31 2014
a(21)-a(23) from Serge Batalov, Feb 17 2015

A082672 Numbers n such that (n! + 2)/2 is a prime.

Original entry on oeis.org

2, 4, 5, 7, 8, 13, 16, 30, 43, 49, 91, 119, 213, 1380, 1637, 2258, 4647, 9701, 12258

Offset: 1

Cino Hilliard, May 18 2003

Cf. A089130.

Cf. n!/m-1 is a prime: A002982, A082671, A139056, A139199-A139205; n!/m+1 is a prime: A002981, A082672, A089085, A139061, A139058, A139063, A139065, A151913, A137390, A139071 (1<=m<=10).

[ n: n in [1..300] | IsPrime((Factorial(n)+2) div 2) ];

Select[Range[10^2], PrimeQ[(#!+2)/2] &] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

\\ x such that (x!+2)/2 is prime
xfactpk(n,k=2) = { for(x=2,n, y = (x!+k)/k; if(isprime(y),print1(x, ", ")) ) }

More terms from Don Reble, Dec 08 2003

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008

A139056 Numbers k for which (k!-3)/3 is prime.

Original entry on oeis.org

4, 6, 12, 16, 29, 34, 43, 111, 137, 181, 528, 2685, 39477, 43697

Offset: 1

Artur Jasinski, Apr 07 2008

Corresponding primes (k!-3)/3 are in A139057.
a(13) > 10000. The PFGW program has been used to certify all the terms up to a(12), using a deterministic test which exploits the factorization of a(n) + 1. - Giovanni Resta, Mar 28 2014
98166 is a member of the sequence but its index is not yet determined. The interval where sieving and tests were not run is [60000,90000]. - Serge Batalov, Feb 24 2015

C. Caldwell. The Prime database entry for the prime generated by a(i)=98166.

Cf. A007749, A117141.
Cf. n!/m-1 is a prime: A002982, A082671, A139056, A139199-A139205.
Cf. n!/m+1 is a prime: A002981, A082672, A089085, A139061, A139058, A139063, A139065, A151913, A137390, A139071.
Cf. m*n!-1 is a prime: A076133, A076134, A099350, A099351, A180627-A180631.
Cf. m*n!+1 is a prime: A051915, A076679-A076683, A178488, A180626, A126896.

a = {}; Do[If[PrimeQ[(-3 + n!)/3], AppendTo[a, n]], {n, 1, 1000}]; a

for(n=1,1000,if(floor(n!/3-1)==n!/3-1,if(ispseudoprime(n!/3-1),print(n)))) \\ Derek Orr, Mar 28 2014

Definition corrected by Derek Orr, Mar 28 2014
a(8)-a(11) from Derek Orr, Mar 28 2014
a(12) from Giovanni Resta, Mar 28 2014
a(13)-a(14) from Serge Batalov, Feb 24 2015

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

A137390 Numbers k for which (9 + k!)/9 is prime.

Original entry on oeis.org

8, 46, 87, 168, 259, 262, 292, 329, 446, 1056, 3562, 11819, 26737

Offset: 1

Artur Jasinski, Apr 09 2008

No other k exists, for k <= 6000. - Dimitris Zygiridis (dmzyg70(AT)gmail.com), Jul 25 2008
The next number in the sequence, if one exists, is greater than 10944. - Robert Price, Mar 16 2010
Borrowing from A139074 another term in this sequence is 26737. There may be others between 10944 and 26737. - Robert Price, Dec 13 2011
There are no other terms for k < 26738. - Robert Price, Feb 10 2012

			a(11) = 3562 because 3562 is the 11th natural number for which k!/9 + 1 is prime. 3562 is the new term.

Cf. A139068 (primes of the form (9 + k!)/9).
Cf. k!/m - 1 is a prime: A002982, A082671, A139056, A139199-A139205.
Cf. (m + k!)/m is a prime: A002981, A082672, A089085, A139061, A139058, A139063, A139065, A151913, A139071.

a = {}; Do[If[PrimeQ[(n! + 9)/9], AppendTo[a, n]], {n, 1, 500}]; a

for(n=6,1e4,if(ispseudoprime(n!/9+1),print1(n", "))) \\ Charles R Greathouse IV, Jul 15 2011

ABC2 $a!/9+1
a: from 6 to 1000 // Jinyuan Wang, Feb 04 2020

Edited by N. J. A. Sloane, May 15 2008 at the suggestion of R. J. Mathar
a(10) corrected from 1053 to 1056 by Dmitry Kamenetsky, Jul 12 2008
a(11) from Dimitris Zygiridis (dmzyg70(AT)gmail.com), Jul 25 2008
a(12)-a(13) from Robert Price, Feb 10 2012

A139058 Numbers n such that (5+n!)/5 is prime.

Original entry on oeis.org

7, 9, 11, 14, 19, 23, 45, 121, 131, 194, 735, 751, 1316, 1372, 2084, 2562, 5678, 5758, 12533, 24222

Offset: 1

Artur Jasinski, Apr 07 2008

nonn

For primes of the form (5+n!)/5 see A139059.

a(21) > 25000. - Robert Price, Nov 20 2016

Cf. A139059.

Cf. n!/m-1 is a prime: A002982, A082671, A139056, A139199-A139205; n!/m+1 is a prime: A002981, A082672, A089085, A139061, A139058, A139063, A139065, A151913, A137390, A139071 (1<=m<=10).

[ n: n in [5..734] | IsPrime((Factorial(n)+5) div 5) ];

a = {}; Do[If[PrimeQ[(n! + 5)/5], AppendTo[a, n]], {n, 1, 751}]; a

A139058(n) = local(k=(n!+5)\5); if(isprime(k), k, 0);
for(n=5, 800, if(A139058(n)>0, print1(n, ", ")))

More terms from Serge Batalov, Feb 18 2015

a(19)-a(20) from Robert Price, Nov 20 2016

A139061 Numbers n for which (4+n!)/4 is prime.

Original entry on oeis.org

4, 5, 6, 13, 21, 25, 32, 40, 61, 97, 147, 324, 325, 348, 369, 1290, 1342, 3167, 6612, 8176, 10990

Offset: 1

Artur Jasinski, Apr 07 2008

nonn

For primes of the form (4+k!)/4, see A139060.

a(22) > 25000. - Robert Price, Jan 10 2017

Cf. A082672, A089085, A089130, A117141, A007749, A139056, A139057, A139058, A139059, A139060, A139061, A139061, A139062, A139063, A139064, A139065, A139066, A020458, A139068, A137390, A139070, A139071, A139072.

Cf. n!/m-1 is a prime: A002982, A082671, A139056, A139199-A139205; n!/m+1 is a prime: A002981, A082672, A089085, A139061, A139058, A139063, A139065, A151913, A137390, A139071 (1<=m<=10).

a = {}; Do[If[PrimeQ[(n! + 4)/4], AppendTo[a, n]], {n, 1, 500}]; a
Select[Range[500],PrimeQ[(4+#!)/4]&]  (* Harvey P. Dale, Mar 24 2011 *)

for(n=4,1e3,if(ispseudoprime(n!/4+1),print1(n", "))) \\ Charles R Greathouse IV, Jul 15 2011

More terms from Serge Batalov, Feb 18 2015

a(19) - a(21) from Robert Price, Jan 10 2017

cp's OEIS Frontend

A038507 a(n) = n! + 1.

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

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A002109 Hyperfactorials: Product_{k = 1..n} k^k.

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A089085 Numbers k such that (k! + 3)/3 is prime.

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A082672 Numbers n such that (n! + 2)/2 is a prime.

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A139056 Numbers k for which (k!-3)/3 is prime.

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