A051856 -id:A051856 - cp's OEIS frontend

A084811 Duplicate of A051856.

0, 1, 2, 3, 4, 6, 76

Offset: 1

dead

A087146 Numbers k such that k! + (k+1)! - 1 is prime.

1, 2, 3, 5, 9, 13, 14, 17, 27, 28, 33, 39, 103, 115, 205, 291, 1431, 1532, 1710, 1937, 3901, 3981, 4682, 6569, 20266, 20662

Offset: 1

Farideh Firoozbakht, Aug 19 2003

nonn

291 is in the sequence and also is in the sequence A087147, thus (291!+292!-1,291!+292!+1) is a twin pair of primes. Any additional terms are greater than 1800 with the next prime having more than 5086 digits.
a(27) > 25000. - Robert Price, Jul 12 2015
a(1)-a(26) give certified primes. - Robert Price, Jul 12 2015

			3 is in the sequence because 3!+4!-1=29 is prime.

H. Dubner, Factorial and primorial primes, J. Rec. Math., 19(No. 3, 1987)

Cf. A087147, A051856, A125174 (corresponding primes).

v={}; Do[If[PrimeQ[n!+(n+1)!-1], v=Append[v, n]; Print[v]], {n, 1800}]; v

One more term from Ryan Propper, Aug 13 2005

a(21)-a(26) from Robert Price, Jul 12 2015

A087147 Numbers k such that k! + (k+1)! + 1 is prime.

Original entry on oeis.org

0, 3, 7, 9, 67, 291, 1343, 6984, 12861

Offset: 1

Farideh Firoozbakht, Aug 19 2003

291 is in the sequence and also is in the sequence A087146, thus (291!+292!-1,291!+292!+1) is a twin pair of primes. Any additional terms are greater than 1800 with the next prime having more than 5086 digits.
Next term is greater than 4200. - Gabriel Cunningham (gcasey(AT)mit.edu), Sep 09 2003
a(10) > 25000. - Robert Price, Aug 26 2015
k+1 is not prime because (p-1)! + p! + 1 == 0 mod p and (p-1)! + p! + 1 > p where p is prime. - Seiichi Manyama, Mar 22 2018

			3 is in the sequence because 3!+4!+1=31 is prime.

H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No.3, 1987)

Eric Weisstein's World of Mathematics, Wilson's Theorem

Cf. A087146, A051856.

Primes in A118913. [From Dmitry Kamenetsky, Oct 21 2008]

v={}; Do[If[PrimeQ[n!+(n+1)!+1], v=Append[v, n]; Print[v]], {n, 1800}]; v
Select[Range[0,25000],PrimeQ[#!+(#+1)!+1]&] (* Robert Price, Aug 26 2015 *)

isok(k) = ispseudoprime(k!+(k+1)!+1); \\ Altug Alkan, Mar 22 2018

a(8)-a(9) from Robert Price, Aug 26 2015

A142959 Numbers k such that both (k!)^2 + k! + 1 and (k!)^2 + 1 are prime.

Original entry on oeis.org

1, 2, 3, 4, 76

Offset: 1

Lekraj Beedassy, Jul 13 2008

For the numbers of digits of the associated primes, see A142960.

a(6) > 150000, because A46029(15) > 150000 and A46029(14) = 127162 is not a term here since (127162!)^2 + 127162! + 1 is divisible by 2049703. - Giovanni Resta, Aug 12 2025

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 76, pp 26, Ellipses, Paris 2008.

Cf. A142960, A046029, A051856.

Select[Range[80],With[{c=#!},AllTrue[c^2+{c+1,1},PrimeQ]]&] (* Harvey P. Dale, Aug 11 2025 *)

A046029 INTERSECT A051856. - R. J. Mathar, Aug 08 2008

A051857 Numbers n such that (n!)^2-n!+1 is prime.

Original entry on oeis.org

2, 3, 5, 7, 38, 2319, 2996, 3321, 3892

Offset: 1

Andrew Walker (ajw01(AT)uow.edu.au), Dec 13 1999

a(1)-a(9) are verified primes using BLS option in pfgw. - Robert Price, Aug 24 2014

a(10) > 15000. - Robert Price, Aug 24 2014

C. K. Caldwell, The Prime Pages
C. Nash, Prime Form [?Broken link]
M. Oakes, Re: Gaussian primorial and factorial primes, Primeform, Dec 21 2010
Mike Oakes, Andrew Walker, David Broadhurst, Gaussian primorial and factorial primes, digest of 7 messages in primeform Yahoo group, Dec 20 - Dec 21, 2010.

Cf. A002981, A002982, A046029, A051856.

isok(n) = isprime((n!)^2-n!+1); \\ Michel Marcus, Aug 26 2013

a(7)-a(9) from Robert Price, Aug 24 2014

A162455 Primes of the form (n!)^2 + (n!) + 1.

Original entry on oeis.org

3, 7, 43, 601, 519121

Offset: 1

Daniel Tisdale, Jul 03 2009

nonn

a(6) is associated with 76!=A000142(76)=A051856(7)! and has 223 decimal digits. - R. J. Mathar, Emeric Deutsch and Harvey P. Dale, Jul 14 2009
a(6) =
355509027001074785420251313577077264819432566692554164797700525028005\
008417722668844213916658906516439209129303887999464691915088148300993\
913093163021697465163231343499013584682565554967153792311772997222400\
0000000000000001. - N. J. A. Sloane, Jul 11 2010

a := proc (n) if isprime(factorial(n)^2+factorial(n)+1) = true then factorial(n)^2+factorial(n)+1 else end if end proc: seq(a(n), n = 1 .. 76); # Emeric Deutsch, Jul 21 2009

Select[Table[n!^2+n!+1,{n,150}],PrimeQ] (* Harvey P. Dale, Jul 23 2009 *)
Select[#^2+#+1&/@(Range[150]!),PrimeQ] (* Harvey P. Dale, Nov 24 2024 *)

a(n) = A002061(A000142(A051856(n+1))+1). - R. J. Mathar, Jul 14 2009

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A084811 Duplicate of A051856.

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A087146 Numbers k such that k! + (k+1)! - 1 is prime.

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A087147 Numbers k such that k! + (k+1)! + 1 is prime.

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A142959 Numbers k such that both (k!)^2 + k! + 1 and (k!)^2 + 1 are prime.

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A051857 Numbers n such that (n!)^2-n!+1 is prime.

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A162455 Primes of the form (n!)^2 + (n!) + 1.

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