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A163213 Swinging Wilson remainders ((p-1)$ + (-1)^floor((p+2)/2))/p mod p, p prime. Here '$' denotes the swinging factorial function (A056040).

%I A163213 #12 May 08 2020 17:40:35
%S A163213 1,1,1,3,1,6,9,13,12,2,19,2,5,36,6,19,43,11,47,67,39,41,70,12,17,83,
%T A163213 88,81,25,53,91,97,106,79,43,39,7,29,73,6,79,115
%N A163213 Swinging Wilson remainders ((p-1)$ + (-1)^floor((p+2)/2))/p mod p, p prime. Here '$' denotes the swinging factorial function (A056040).
%C A163213 If this is zero, p is a swinging Wilson prime.
%H A163213 G. C. Greubel, <a href="/A163213/b163213.txt">Table of n, a(n) for n = 1..1000</a>
%H A163213 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H A163213 Peter Luschny, <a href="http://www.luschny.de/math/primes/SwingingPrimes.html"> Swinging Primes.</a>
%e A163213 The swinging Wilson quotient related to the 5th prime is (252+1)/11=23, so the 5th term is 23 mod 11 = 1.
%p A163213 WR := proc(f,r,n) map(p->(f(p-1)+r(p))/p mod p,select(isprime,[$1..n])) end:
%p A163213 A002068 := n -> WR(factorial,p->1,n);
%p A163213 A163213 := n -> WR(swing,p->(-1)^iquo(p+2,2),n);
%t A163213 sf[n_] := n!/Quotient[n, 2]!^2; a[n_] := (p = Prime[n]; Mod[(sf[p - 1] + (-1)^Floor[(p + 2)/2])/p, p]); Table[a[n], {n, 1, 42}] (* _Jean-François Alcover_, Jun 28 2013 *)
%o A163213 (PARI) sf(n)=n!/(n\2)!^2
%o A163213 apply(p->sf(p-1)\/p%p, primes(100)) \\ _Charles R Greathouse IV_, Dec 11 2016
%Y A163213 Cf. A163211, A002068, A163210.
%K A163213 nonn
%O A163213 1,4
%A A163213 _Peter Luschny_, Jul 24 2009