A002981 -id:A002981 - cp's OEIS frontend

A049433 Numbers k such that k! - (k-1)! - 1 is prime.

3, 4, 6, 8, 9, 12, 28, 78, 99, 184, 286, 291, 398, 411, 600, 718, 732, 889, 1963, 2240, 2242, 2533, 8800, 11403, 18335, 20277, 21029

Offset: 1

Paul Jobling (paul.jobling(AT)whitecross.com)

nonn

There is no further term up to 1400. - Farideh Firoozbakht, Jul 18 2003

a(25) > 12000. [Donovan Johnson, Dec 18 2009]

			6 is a term since 6! - (6-1)! - 1 = 599 is prime.

Index entries for sequences related to factorial numbers

Cf. A049432, A049985, A090704.

Cf. A001272, A002981, A002982.

a(n) = A090704(n) + 1 = n*n! + 1 = ((n+1)-1)*n! + 1 = (n+1)! - n! + 1 .

More terms from Farideh Firoozbakht, Jul 18 2003
Corrected offset, edited definition and a(19)-a(24) from Donovan Johnson, Dec 18 2009
a(25)-a(27) from Michael S. Branicky, Jun 13 2025

A064144 a(n) is the number of divisors of n!+1.

Original entry on oeis.org

2, 2, 2, 3, 3, 4, 3, 4, 8, 4, 2, 6, 4, 4, 8, 32, 8, 64, 4, 4, 8, 8, 12, 4, 4, 4, 2, 4, 8, 32, 16, 16, 32, 4, 32, 64, 2, 4, 16, 128, 2, 8, 16, 8, 8, 8, 16, 4, 32, 32, 64, 16, 16, 4, 4, 16, 8, 16, 4, 16, 16, 8, 32, 8

Offset: 1

Vladeta Jovovic, Sep 11 2001

nonn

Max Alekseyev, Table of n, a(n) for n = 1..139
William Gerst, A conjecture on the prime factorization of n!+1, arXiv:1809.07360 [math.GM], 2018.
Hisanori Mishima, Factorizations of many number sequences

Cf. A000005, A002583, A027423, A002981, A002982, A064145, A078778, A181764.

Do[ Print[ DivisorSigma[0, n! + 1]], {n, 1, 40} ]

a(n) = numdiv(n! + 1); \\ Harry J. Smith, Sep 09 2009

from math import factorial
from sympy import divisor_count
def A064144(n): return divisor_count(factorial(n)+1) # Chai Wah Wu, Oct 20 2023

a(n) = tau(n!+1).

More terms from Robert G. Wilson v, Oct 04 2001
a(42)-a(64) from Harry J. Smith, Sep 09 2009
Edited by Jon E. Schoenfield, Jun 21 2018

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

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

A064164 EHS numbers: k such that there is a prime p satisfying k! + 1 == 0 (mod p) and p !== 1 (mod k).

Original entry on oeis.org

8, 9, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

R. K. Guy, Sep 20 2001

The complement of this sequence (A064295) is a superset of A002981, that is, terms of A002981 do not appear in this sequence.

Hardy & Subbarao prove that this sequence is infinite, see their Theorem 2.12. - Charles R Greathouse IV, Sep 10 2015

Amiram Eldar, Table of n, a(n) for n = 1..120
G. E. Hardy and M. V. Subbarao, A modified problem of Pillai and some related questions, Amer. Math. Monthly 109:6 (2002), pp. 554-559.
H. Mishima, Factors of N!+1

The smallest associated primes p are given in A064229.

Cf. A002981, A064295.

Do[k = 1; While[p = Prime[k]; k < 10^8 && Not[ Nor[ Mod[n! + 1, p] != 0, Mod[p, n] == 1]], k++ ]; If[k != 10^8, Print[n, " ", p]], {n, 2, 88}]

is(n)=my(f=factor(n!+1)[,1]); for(i=1,#f, if(f[i]%n != 1, return(n>1))); 0 \\ Charles R Greathouse IV, Sep 10 2015

Corrected and extended by Don Reble, Sep 23 2001

A084749 Numbers m such that m! + p is a prime, where p is the smallest prime > m.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 10, 33, 44, 48, 52, 64, 73, 92, 119, 182, 487, 603, 987, 4884, 6822, 8070, 11079, 13659, 17659

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 16 2003

Next term, if it exists, is >4800. - Ryan Propper, Jan 02 2007
From Farideh Firoozbakht, Oct 21 2009: (Start)
Numbers corresponding to a(19)-a(24) are probable primes.
There is no further term up to 8300. (End)

			727 = 6! + 7 is a prime but 8! + 11 is composite hence 6 is a member but 8 is not.
7 is in the sequence because 7!=5040, nextprime(7)=11 and 5040+11 is prime.

Cf. A084748, A084750, A092028, A064278, A002981, A151892.

Do[If[PrimeQ[k!+NextPrime[k]], Print[k]], {k, 0, 1525}] (* Farideh Firoozbakht, Feb 26 2004 *)
Select[Range[0,500],PrimeQ[#!+NextPrime[#]]&] (* The program generates the first 19 terms of the sequence. *) (* Harvey P. Dale, Jul 16 2025 *)

More terms from Farideh Firoozbakht, Feb 26 2004
Edited by N. J. A. Sloane at the suggestion of Artur Jasinski, Apr 14 2008
a(22)-a(24) from Farideh Firoozbakht, Oct 21 2009
a(25) from Michael S. Branicky, Aug 05 2024
a(26)-a(27) from Michael S. Branicky, May 25 2025

A084846 mu(n!+1), where mu is the Moebius function (A008683).

Original entry on oeis.org

-1, -1, -1, -1, 0, 0, 1, 0, 1, -1, 1, -1, 0, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, 0, 1, 1, 1, -1, 1, -1, -1, 1, 1, -1, 1, -1, 1, -1, 1, 1, -1, -1, -1, 1, -1, -1, -1, 1, 1, -1, -1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, 1, -1, 1, -1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, -1, -1, -1, -1, -1, -1, 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1

Offset: 0

Rick L. Shepherd, Jun 10 2003

			a(6)=1 because 6!+1 = 721 = 7 * 103, the product of two different primes and thus mu(6!+1) = (-1)^2 = 1.

Amiram Eldar, Table of n, a(n) for n = 0..139
Paul Leyland, Factors of n!+1.
Markus Tervooren et al., Factorization of n!+1, FactorDB.

Cf. A008683 (mu(n)), A054990 (bigomega(n!+1)), A066856 (omega(n!+1)), A064237 (n!+1 divisible by a square), A002981 (n!+1 is prime).

[MoebiusMu(Factorial(n)+1) : n in [1..45]];

MoebiusMu[Range[0, 50]! + 1] (* Paolo Xausa, Feb 07 2025 *)

PARI
```
for(n=0,45,print1(moebius(n!+1),","))
```

If n is in A064237, then a(n) = 0. Otherwise a(n) = (-1)^A054990(n) = (-1)^A066856(n). - Max Alekseyev, Oct 08 2019

a(112) corrected, a(113)-a(114) added by Max Alekseyev, May 28 2015

a(106)-a(107) corrected by Amiram Eldar, Oct 03 2019

A090660 Numbers n such that n*nextprime((n-1)!)-nextprime(n!) < 0.

Original entry on oeis.org

3, 4, 12, 28, 38, 42, 74, 78, 117, 155, 321, 341, 400, 428, 873, 1478, 6381, 26952

Offset: 1

Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Dec 15 2003

nonn

3*nextprime((3-1)!) - nextprime(3!) = 3*nextprime(2!) - nextprime(3!) = 3*2 - 7 = -1.

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.

Cf. A090661, A089014.

Equals A002981 + 1.

NextPrim[ n_ ] := Block[ {k = n + 1}, While[ !PrimeQ[ k ], k++ ]; k ]; Select[ Range[ 260 ], #*NextPrim[ (# - 1)! ] - NextPrim[ #! ] < 0 & ] (* Robert G. Wilson v *)

Better description from Don Reble, Dec 20 2003
Three more terms from Robert G. Wilson v, Dec 20 2003
a(14) from Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Jan 05 2004

A151892 Numbers m such that m! + (next prime after m!) is prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 14, 15, 20, 25, 32, 34, 35, 67, 89, 191, 316, 411, 1213, 1280, 2022, 2267

Offset: 1

Artur Jasinski, Apr 12 2008

Cf. A002981, A151894, A151893, A151903, A084749.

a = {}; Do[If[PrimeQ[n! + NextPrime[n! ]], AppendTo[a, n]], {n, 200}]; a (* Artur Jasinski *)
Select[Range[420],PrimeQ[#!+NextPrime[#!]]&] (* Harvey P. Dale, Aug 20 2021 *)

a(17)-a(18) from Robert G. Wilson v, Jun 11 2010
a(19)-a(20) from Michael S. Branicky, May 27 2023
a(21)-a(22) from Michael S. Branicky, Aug 03 2024

A151893 a(n) = smallest number k such that n! + k-th prime after n! is prime.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 4, 10, 2, 4, 3, 1, 1, 4, 2, 9, 14, 1, 6, 14, 6, 5, 1, 11, 3, 35, 14, 20, 4, 1, 10, 1, 1, 6, 37, 33, 25, 17, 62, 2, 5, 26, 12, 10, 11, 37, 9, 9, 4, 50, 32, 9, 9, 7, 9, 47, 10, 40, 80, 60, 3, 18, 6, 2, 1

Offset: 1

Artur Jasinski, Apr 12 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..300

Cf. A002981, A151892, A151894, A151903, A139213, A139214.

a = {}; Do[k = 1; While[ ! PrimeQ[n! + NextPrime[n!, k]], k++ ]; Print[k]; AppendTo[a, k], {n, 1, 200}]; a

cp's OEIS Frontend

A049433 Numbers k such that k! - (k-1)! - 1 is prime.

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A064144 a(n) is the number of divisors of n!+1.

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

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

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Maple

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A064164 EHS numbers: k such that there is a prime p satisfying k! + 1 == 0 (mod p) and p !== 1 (mod k).

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A084749 Numbers m such that m! + p is a prime, where p is the smallest prime > m.

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A084846 mu(n!+1), where mu is the Moebius function (A008683).

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Magma

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