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

1,4

Comments

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

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.

Example

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

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 *)

Prog

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

Crossrefs

Cf. A163211, A002068, A163210.

Sequence in context: A065918 A362695 A020861 * A095066 A169955 A084536

Adjacent sequences: A163210 A163211 A163212 * A163214 A163215 A163216

Keyword

nonn

Author

Peter Luschny, Jul 24 2009

Status

approved