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

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

Offset

1,2

Comments

Except for the first term, all terms are primes because for n > 1, n divides (n-1)! + 1 iff n is prime. There are no other terms up to 6550 and the corresponding next prime has more than 22150 digits.

No more terms below 30941.

Links

Table of n, a(n) for n=1..8.

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

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

Example

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

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 *)

Prog

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

Crossrefs

Sequence in context: A375096 A031134 A144231 * A092029 A259564 A308764

Adjacent sequences: A050296 A050297 A050298 * A050300 A050301 A050302

Keyword

nonn,more,changed

Author

N. J. A. Sloane, Apr 09 2003

Extensions

1321 and 2621 from Mike Oakes, Aug 20 2003

Additional comments from Farideh Firoozbakht, Mar 19 2004

Status

approved