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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).
4
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
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
KEYWORD
nonn
AUTHOR
Peter Luschny, Jul 24 2009
STATUS
approved