A002981 -id:A002981 - cp's OEIS frontend

A139063 Numbers k for which (6+k!)/6 is prime.

3, 4, 10, 11, 13, 14, 17, 21, 82, 115, 165, 167, 173, 174, 208, 225, 380, 655, 1187, 2000, 2568, 3010, 4542, 8750, 12257, 12601, 24083

Offset: 1

Artur Jasinski, Apr 07 2008

nonn

For primes of the form (6+k!)/6, see A139062.

a(28) > 25000. - Robert Price, Nov 20 2016

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

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! + 6)/6], AppendTo[a, n]], {n, 1, 500}]; a

for(k=3,1e3,if(ispseudoprime(k!/6+1),print1(k", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(18) and a(19) from Robert Israel, May 19 2014
More terms from Serge Batalov, Feb 18 2015
a(24)-a(27) from Robert Price, Nov 20 2016

A139065 Numbers k for which (7+k!)/7 is prime.

Original entry on oeis.org

11, 15, 16, 25, 35, 59, 64, 68, 82, 121, 149, 238, 584, 912, 3349, 4111, 4324, 15314, 19944, 20658, 22740, 23364

Offset: 1

Artur Jasinski, Apr 07 2008

For primes of the form (7+k!)/7, see A139064.

a(23) > 25000. - Robert Price, Nov 20 2016

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

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! + 7)/7], AppendTo[a, n]], {n, 1, 500}]; a
Select[Range[500],PrimeQ[(7+#!)/7]&] (* Harvey P. Dale, Sep 01 2014 *)

for(k=7,1e3,if(ispseudoprime(k!/7+1),print1(k", "))) \\ Charles R Greathouse IV, Jul 15 2011

More terms from Serge Batalov, Feb 18 2015

a(18)-a(22) from Robert Price, Nov 20 2016

A033932 Least positive m such that n! + m is prime.

Original entry on oeis.org

1, 1, 1, 1, 5, 7, 7, 11, 23, 17, 11, 1, 29, 67, 19, 43, 23, 31, 37, 89, 29, 31, 31, 97, 131, 41, 59, 1, 67, 223, 107, 127, 79, 37, 97, 61, 131, 1, 43, 97, 53, 1, 97, 71, 47, 239, 101, 233, 53, 83, 61, 271, 53, 71, 223, 71, 149, 107, 283, 293, 271, 769, 131, 271

Offset: 0

Jeff Burch

Conjecture: No term is a composite number. a(n) is a prime > 3*prime(k), where k is such that prime(k) < n <= prime(k+1). - Amarnath Murthy, Apr 07 2004

Terms after n = 2000 in the b-file correspond to Fermat and Lucas PRP. - Phillip Poplin, Oct 12 2019

Phillip Poplin, Table of n, a(n) for n = 0..4000 (first 501 terms from T. D. Noe, then up to n=2000 from Hans Havermann)
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.
Index entries for sequences related to factorial numbers

Cf. A000142, A002981, A033933, A037151, A037153, A056752, A053714, A151800.

a:= n-> (f-> nextprime(f)-f)(n!):
seq(a(n), n=0..70);  # Alois P. Heinz, Feb 22 2023

a[n_] := (an = 1; While[ !PrimeQ[n! + an], an++]; an); Table[a[n], {n, 0, 63}] (* Jean-François Alcover, Dec 05 2012 *)
NextPrime[#]-#&/@(Range[0,70]!) (* Harvey P. Dale, May 18 2014 *)

for(n=0,70, k=1; while(!isprime(n!+k), k++); print1(k,","))

a(n) = nextprime(n!+1) - n!; \\ Michel Marcus, Dec 25 2020

from sympy import factorial, nextprime
def a(n): fn = factorial(n); return nextprime(fn) - fn
print([a(n) for n in range(64)]) # Michael S. Branicky, May 22 2022

a(n) = A151800(n!) - n!. - Max Alekseyev, Jul 23 2014

More terms from Jud McCranie
a(21) onwards from Wouter Meeussen
Better description from Rick L. Shepherd, Nov 06 2002

A005234 Primes p such that 1 + product of primes up to p is prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, 42209, 145823, 366439, 392113, 4328927, 5256037, 6369619, 7351117, 9562633

Offset: 1

N. J. A. Sloane

Conjecture: if p# + 1 is a prime number, then the next prime is less than p# + exp(1)*p. - Arkadiusz Wesolowski, Feb 20 2013

Conjecture: if p# + 1 is a prime, then the next prime is less than p# + p^2. - Thomas Ordowski, Apr 07 2013

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 134.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 211, p. 61, Ellipses, Paris 2008.
H. Dubner, A new primorial prime, J. Rec. Math., 21 (No. 4, 1989), 276.
R. K. Guy, Unsolved Problems in Number Theory, Section A2.
F. Le Lionnais, Les Nombres Remarquables, Paris, Hermann, 1983, p. 109, 1983.
Paulo Ribenboim, The New Book of Prime Number Records, p. 13.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 4-5.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 112.

C. K. Caldwell, Prime Pages: Database Search.
C. K. Caldwell, Primorial Primes.
C. K. Caldwell and Y. Gallot, On the primality of n!+-1 and 2*3*5*...*p+-1, Math. Comp., 71 (2001), 441-448.
H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3, 1987), 197-203. (Annotated scanned copy)
H. Dubner, A new primorial prime, J. Rec. Math., 21.4 (1989), 276. (Annotated scanned copy)
H. Dubner and N. J. A. Sloane, Correspondence, 1991.
Des MacHale, Infinitely many proofs that there are infinitely many primes, Math. Gazette, 97 (No. 540, 2013), 495-498.
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.
R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations.
Eric Weisstein's World of Mathematics, Euclid Number.
Eric Weisstein's World of Mathematics, Primorial Prime.

Cf. A006862 (Euclid numbers).
Cf. A014545 (Primorial plus 1 prime indices: n such that 1 + (Product of first n primes) is prime).
Cf. A018239 (Primorial plus 1 primes).
Cf. A002110, A006794, A057704.

[p:p in PrimesUpTo(3000)|IsPrime(&*PrimesUpTo(p)+1)]; // Marius A. Burtea, Mar 25 2019

N:= 5000: # to get all terms <= N
Primes:= select(isprime, [$2..N]):
P:= 1: count:= 0:
for n from 1 to nops(Primes) do
   P:= P*Primes[n];
   if isprime(P+1) then
     count:= count+1; A[count]:= Primes[n]
   fi
od:
seq(A[i],i=1..count); # Robert Israel, Nov 03 2015

(* This program is not convenient for large values of p *) p = pp = 1; Reap[While[p < 5000, p = NextPrime[p]; pp = pp*p; If[PrimeQ[1 + pp], Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Dec 31 2012 *)
With[{p = Prime[Range[200]]}, p[[Flatten[Position[Rest[FoldList[Times, 1, p]] + 1, ?PrimeQ]]]]] (* _Eric W. Weisstein, Nov 03 2015 *)

is(n)=isprime(n) && ispseudoprime(prod(i=1,primepi(n),prime(i))+1) \\ Charles R Greathouse IV, Feb 20 2013

is(n)=isprime(n) && ispseudoprime(factorback(primes([2,n]))+1) \\ M. F. Hasler, May 31 2018

a(n) = A000040(A014545(n+1)). - M. F. Hasler, May 31 2018

42209 sent in by Chris Nash (chrisnash(AT)cwix.com).
145823 discovered and sent in by Arlin Anderson (starship1(AT)gmail.com) and Don Robinson (donald.robinson(AT)itt.com), Jun 01 2000
366439, 392113 from Eric W. Weisstein, Mar 13 2004 (based on information in A014545)
a(23) from Jeppe Stig Nielsen, Aug 08 2024
a(24) from Jeppe Stig Nielsen, Sep 01 2024
a(25) from Jeppe Stig Nielsen, Sep 24 2024
a(26) from Jeppe Stig Nielsen, Nov 10 2024
a(27) from Jeppe Stig Nielsen, Aug 21 2025

A139071 Numbers k for which (10+k!)/10 is prime.

Original entry on oeis.org

5, 6, 11, 12, 15, 23, 26, 37, 45, 108, 112, 129, 137, 148, 172, 248, 760, 807, 975, 1398, 5231, 8765, 24182

Offset: 1

Artur Jasinski, Apr 07 2008

nonn

Primes of the form (10+k!)/10 see A139070.

a(24) > 25000. - Robert Price, Nov 08 2016

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! + 10)/10], AppendTo[a, n]], {n, 1, 500}]; a

for(k=5,1e3,if(ispseudoprime(k!/10+1),print1(k", "))) \\ Charles R Greathouse IV, Jul 15 2011

More terms from Serge Batalov, Feb 18 2015

a(22)-a(23) from Robert Price, Nov 08 2016

A082671 Numbers n such that (n!-2)/2 is a prime.

Original entry on oeis.org

3, 4, 5, 6, 9, 31, 41, 373, 589, 812, 989, 1115, 1488, 1864, 1918, 4412, 4686, 5821, 13830

Offset: 1

Cino Hilliard, May 18 2003

			(4!-2)/2 = 11 is a prime.

Cf. n!/m-1 is a prime: A002982, A082671, A139056, A139199, A139200, A139201, A139202, A139203, A139204, A139205; n!/m+1 is a prime: A002981, A082672, A089085, A139061, A139058, A139063, A139065, A151913, A137390, A139071

[n: n in [1..600]| IsPrime((Factorial(n)-2) div 2)]; // Vincenzo Librandi, Feb 18 2015

Select[Range[0, 14000], PrimeQ[(#! - 2) / 2] &] (* Vincenzo Librandi, Feb 18 2015 *)

xfactpk(n,k=2) = { for(x=2,n, y = (x!-k)/k; if(isprime(y),print1(x", ")) ) }

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

Edited by T. D. Noe, Oct 30 2008

A088332 Primes of the form k! + 1.

Original entry on oeis.org

2, 3, 7, 39916801, 10888869450418352160768000001, 13763753091226345046315979581580902400000001, 33452526613163807108170062053440751665152000000001

Offset: 1

Cino Hilliard, Nov 06 2003

nonn

The next term is too large to include.
Of course 2 = 0! + 1 = 1! + 1 has two such representations.
Prime numbers that are the sum of two factorial numbers. - Juri-Stepan Gerasimov, Nov 08 2010

			3! + 1 = 7 is prime.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 118.

Amiram Eldar, Table of n, a(n) for n = 1..15 (terms 1..11 from T. D. Noe)
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], 2012-2023. - From _N. J. A. Sloane_, Jun 13 2012

Cf. A002981 (values of k), A038507, A062701.

lst={};Do[p=n!+1;If[PrimeQ[p],AppendTo[lst,p]],{n,0,3*5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)
Select[Range[50]!+1,PrimeQ] (* Harvey P. Dale, May 17 2025 *)

factp1prime(n)=for(x=1,n,xf=x!+1; if(isprime(xf),print1(xf",")))

a(n) = A038507(A002981(n+1)). - Elmo R. Oliveira, Apr 16 2025

A139199 Numbers k such that (k!-4)/4 is prime.

Original entry on oeis.org

4, 5, 6, 7, 8, 10, 15, 18, 23, 157, 165, 183, 184, 362, 611, 908, 2940, 6875, 9446, 16041

Offset: 1

Artur Jasinski, Apr 11 2008

Numbers k such that (k!-m)/m is prime:
for m=1 see A002982
for m=2 prime or pseudoprime see A082671
for m=3 see A139056
for m=4 see A139199
for m=5 see A139200
for m=6 see A139201
for m=7 see A139202
for m=8 see A139203
for m=9 see A139204
for m=10 see A139205
a(17) > 2000 - Ray G. Opao, Sep 30 2008
a(21) > 25000 - Robert Price, Sep 25 2016

Cf. A139189, A139190, A139191, A139192, A139193, A139194, A139195, A139196, A139197, A139198.

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], Print[a]; AppendTo[a, n]], {n, 1, 184}]; a (*Artur Jasinski*)

is(n)=n>3 && isprime(n!/4-1) \\ Charles R Greathouse IV, Apr 29 2015

a(15)-a(16) from Ray G. Opao, Sep 30 2008
a(17) from Serge Batalov, Feb 18 2015
a(18)-a(20) from Robert Price, Sep 25 2016

A139205 Numbers k such that (k!-10)/10 is prime.

Original entry on oeis.org

5, 6, 7, 11, 13, 17, 28, 81, 87, 433, 640, 647, 798, 1026, 1216, 1277, 3825, 6684

Offset: 1

Artur Jasinski, Apr 11 2008

a(19) > 25000. - Robert Price, Dec 23 2016

Cf. A139189, A139190, A139191, A139192, A139193, A139194, A139195, A139196, A139197, A139198.

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! - 10)/10], Print[a]; AppendTo[a, n]], {n, 1, 300}]; a (*Artur Jasinski*)
Select[Range[700],PrimeQ[(#!-10)/10]&] (* Harvey P. Dale, Feb 15 2015 *)

One additional term (a(12)) from Harvey P. Dale, Feb 15 2015
More terms from Serge Batalov, Feb 18 2015
a(18) from Robert Price, Dec 23 2016

A151913 Numbers n for which (8+n!)/8 is prime.

Original entry on oeis.org

7, 9, 10, 12, 14, 20, 23, 24, 29, 44, 108, 2049, 3072, 4862, 8807, 15089

Offset: 1

Artur Jasinski, Apr 07 2008

nonn

a(17) > 25000. - Robert Price, Dec 20 2016

For primes of the form (8+k!!)/8 see A139066.
Cf. A089130, A117141, A139035, A139057, A139059, A139060, A139062, A139064, A139067.
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! + 8)/8], AppendTo[a, n]], {n, 1, 500}]; a

is(n)=n>6 && isprime((8+n!)/8) \\ Charles R Greathouse IV, Apr 29 2016

Definition corrected Feb 24 2010
More terms from Serge Batalov, Feb 18 2015
a(15)-a(16) from Robert Price, Dec 20 2016

cp's OEIS Frontend

A139063 Numbers k for which (6+k!)/6 is prime.

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A139065 Numbers k for which (7+k!)/7 is prime.

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A033932 Least positive m such that n! + m is prime.

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A005234 Primes p such that 1 + product of primes up to p is prime.

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A139071 Numbers k for which (10+k!)/10 is prime.

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A082671 Numbers n such that (n!-2)/2 is a prime.

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A088332 Primes of the form k! + 1.

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