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

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, 88, 81, 25, 53, 91, 97, 106, 79, 43, 39, 7, 29, 73, 6, 79, 115

Offset: 1

Author

Peter Luschny, Jul 24 2009

Keywords

• nonn

Comments

If this is zero, p is a swinging Wilson prime.

Examples

			The swinging Wilson quotient related to the 5th prime is (252+1)/11=23, so the 5th term is 23 mod 11 = 1.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
• Peter Luschny, Swinging Primes.

Crossrefs

Cf. A163211, A002068, A163210.

Programs

Maple

WR := proc(f,r,n) map(p->(f(p-1)+r(p))/p mod p,select(isprime,[$1..n])) end:
A002068 := n -> WR(factorial,p->1,n);
A163213 := n -> WR(swing,p->(-1)^iquo(p+2,2),n);

Mathematica

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 *)

PARI

sf(n)=n!/(n\2)!^2
apply(p->sf(p-1)\/p%p, primes(100)) \\ Charles R Greathouse IV, Dec 11 2016