|
%I #34 May 08 2020 17:38:33
%S 5,103,329891,10513391193507374500051862069
%N Wilson quotients (A007619) which are primes.
%C a(5) = A007619(137), a(6) = A007619(216), a(7) = A007619(381).
%C Same as A122696 without its initial term 2. - _Jonathan Sondow_, May 19 2013
%H Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H Peter Luschny, <a href="http://www.luschny.de/math/primes/SwingingPrimes.html"> Swinging Primes.</a>
%H J. Sondow, <a href="http://arxiv.org/abs/1110.3113"> Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771</a>, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.
%H J. Sondow, <a href="https://doi.org/10.1007/978-1-4939-1601-6_17">Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771</a>, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.
%F a(n) = A122696(n+1) = A007619(A225906(n)) = ((A050299(n+1)-1)!+1)/A050299(n+1). - _Jonathan Sondow_, May 19 2013
%e The quotient (720+1)/7 = 103 is a Wilson quotient and a prime, so 103 is a member.
%p # WQ defined in A163210.
%p A163212 := n -> select(isprime,WQ(factorial,p->1,n)):
%t Select[Table[p = Prime[n]; ((p-1)!+1)/p, {n, 1, 15}], PrimeQ] (* _Jean-François Alcover_, Jun 28 2013 *)
%o (PARI) forprime(p=2, 1e4, a=((p-1)!+1)/p; if(ispseudoprime(a), print1(a, ", "))) \\ _Felix Fröhlich_, Aug 03 2014
%Y Cf. A050299, A163211, A007619, A122696, A163210, A163213, A163209, A225906.
%K nonn
%O 1,1
%A _Peter Luschny_, Jul 24 2009
|
