A122696 Primes of the form ((k-1)! + 1)/k.

2, 5, 103, 329891, 10513391193507374500051862069

Offset: 1

Author

Alexander Adamchuk, Sep 22 2006

Keywords

• nonn

Comments

A163212, Wilson quotients (A007619: ((p-1)!+1)/p) which are primes, is a subsequence. Corresponding numbers n such that ((n-1)! + 1)/n is prime are listed in A050299 = {1, 5, 7, 11, 29, 773, 1321, 2621, ...}. a(6) has 1893 digits. a(7) has 3545 digits. a(8) has 7817 digits.

Except for a(1) = 2, same as A163212. - Jonathan Sondow, May 20 2013

Links

• 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.

Crossrefs

A050299 is the main entry for this sequence.

Cf. A007619, A163212.

Programs

Mathematica

Select[Table[((k-1)!+1)/k,{k,30}],PrimeQ] (* James C. McMahon, Nov 09 2024 *)

PARI

is(n)=isprime(((n-1)!+1)/n) \\ Anders Hellström, Nov 22 2015 \\ This program actually produces A050299 - Michel Marcus, Aug 02 2016

PARI

for(n=1, 1e2, if(((n-1)!+1)%n==0 && isprime(k=((n-1)!+1)/n), print1(k, ", "))) \\ Altug Alkan, Nov 22 2015

Formula

a(n) = A163212(n-1) = ((A050299(n)-1)! + 1)/A050299(n). - Jonathan Sondow, May 19 2013

Extensions

The next term is too large to include.

a(4) and first comment corrected by Gionata Neri, Aug 02 2016