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Wilson quotients (A007619) which are primes.
5

%I #40 Nov 08 2024 07:14:54

%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 Jonathan 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 Jonathan 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,more

%O 1,1

%A _Peter Luschny_, Jul 24 2009