A050299 Numbers k such that ((k-1)! + 1)/k is prime.

1, 5, 7, 11, 29, 773, 1321, 2621

Offset: 1

Author

N. J. A. Sloane, Apr 09 2003

Keywords

• nonn
• more

Comments

Except for the first term, all terms are primes because for n > 1, n divides (n-1)! + 1 iff n is prime. - Farideh Firoozbakht, Mar 19 2004

a(9) >= 30941.

Examples

			7 is in the sequence because (6!+1)/7=103 is prime.
		

Links

• Mike Oakes, posting to Number Theory List, Aug 20 2003
• Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.
• Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.
• Javier Soria, posting to Number Theory List, Apr 08 2003

Programs

Mathematica

v={1};Do[If[PrimeQ[((Prime[n]-1)!+1)/Prime[n]], v=Append[v, Prime[n]];Print[v]], {n, 845}]
Select[Range[2630],PrimeQ[((#-1)!+1)/#]&] (* Harvey P. Dale, Aug 18 2024 *)

PARI

is(n)=((n-1)!+1)%n==0 && isprime(((n-1)!+1)/n) \\ Anders Hellström, Nov 22 2015

Formula

((a(n)-1)! + 1)/a(n) = A122696(n) = A007619(A000720(A050299(n))) for n > 1. - Jonathan Sondow, Aug 07 2011

a(n) = prime(A225906(n-1)) for n > 1. - Jonathan Sondow, May 20 2013

Extensions

a(7)-a(8) from Mike Oakes, Aug 20 2003