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%I #16 May 08 2020 17:40:14
%S 2,3,5,31,67,139,631,9743,16253,17977,27901,37589
%N Primes p such that p$ + 1 is also prime. Here '$' denotes the swinging factorial function (A056040).
%C a(n) are the primes in A163077.
%H Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H Peter Luschny, <a href="http://www.luschny.de/math/primes/SwingingPrimes.html"> Swinging Primes.</a>
%e 5 is prime and 5$ + 1 = 30 + 1 = 31 is prime, so 5 is in the sequence.
%p a := proc(n) select(isprime,select(k -> isprime(A056040(k)+1),[$0..n])) end:
%t f[n_] := 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]; p = 2; lst = {}; While[p < 38000, a = f@p + 1; If[ PrimeQ@a, AppendTo[ lst, p]; Print@p]; p = NextPrime@p]; lst (* _Robert G. Wilson v_, Aug 08 2010 *)
%o (PARI) is(k) = isprime(k) && ispseudoprime(1+k!/(k\2)!^2); \\ _Jinyuan Wang_, Mar 22 2020
%Y Cf. A002981, A062363, A093804, A163077.
%Y Cf. A056040. - _Robert G. Wilson v_, Aug 09 2010
%K nonn,more
%O 1,1
%A _Peter Luschny_, Jul 21 2009
%E a(8)-a(12) from _Robert G. Wilson v_, Aug 08 2010