A050299
Numbers k such that ((k-1)! + 1)/k is prime.
Original entry on oeis.org
1, 5, 7, 11, 29, 773, 1321, 2621
Offset: 1
7 is in the sequence because (6!+1)/7=103 is prime.
- 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
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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 *)
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is(n)=((n-1)!+1)%n==0 && isprime(((n-1)!+1)/n) \\ Anders Hellström, Nov 22 2015
A163212
Wilson quotients (A007619) which are primes.
Original entry on oeis.org
5, 103, 329891, 10513391193507374500051862069
Offset: 1
The quotient (720+1)/7 = 103 is a Wilson quotient and a prime, so 103 is a member.
- Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
- Peter Luschny, Swinging Primes.
- 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.
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# WQ defined in A163210.
A163212 := n -> select(isprime,WQ(factorial,p->1,n)):
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Select[Table[p = Prime[n]; ((p-1)!+1)/p, {n, 1, 15}], PrimeQ] (* Jean-François Alcover, Jun 28 2013 *)
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forprime(p=2, 1e4, a=((p-1)!+1)/p; if(ispseudoprime(a), print1(a, ", "))) \\ Felix Fröhlich, Aug 03 2014
A225906
Indices of primes whose Wilson quotients are also prime.
Original entry on oeis.org
3, 4, 5, 10, 137, 216, 381
Offset: 1
The Wilson quotient of 7 is ((7-1)!+1)/7 = 103, which is prime, and 7 is the 4th prime, so 4 is a member.
- J. 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
- J. 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.
A225672
Primes p such that the Wilson quotient of the p-th prime is also prime.
Original entry on oeis.org
The 5th prime is 11 and the Wilson quotient of 11 is ((11-1)!+1)/11 = 329891, which is prime, so 5 is a term.
- 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
- 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.
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