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

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

Offset: 1

Author

Farideh Firoozbakht, Aug 19 2003

Keywords

• more
• nonn

Comments

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

Examples

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

References

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

Links

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

Crossrefs

Cf. A087146, A051856.

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

Programs

Mathematica

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 *)

PARI

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

Extensions

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