A163210 Swinging Wilson quotients ((p-1)$ +(-1)^floor((p+2)/2))/p, p prime. Here '$' denotes the swinging factorial function (A056040).

1, 1, 1, 3, 23, 71, 757, 2559, 30671, 1383331, 5003791, 245273927, 3362110459, 12517624987, 175179377183, 9356953451851, 509614686432899, 1938763632210843, 107752663194272623

Offset

1,4

Links

G. C. Greubel, Table of n, a(n) for n = 1..470

M. E. Bassett, S. Majid, Finite noncommutative geometries related to F_p[x], arXiv:1603.00426 [math.QA], 2016.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Peter Luschny, Swinging Primes.

Example

The 5th prime is 11, (11-1)$ = 252, the remainder term is (-1)^floor((11+2)/2)=1. So the quotient (252+1)/11 = 23 is the 5th member of the sequence.

Maple

swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
WQ := proc(f, r, n) map(p->(f(p-1)+r(p))/p, select(isprime, [$1..n])) end:
A163210 := n -> WQ(swing, p->(-1)^iquo(p+2, 2), n);

Mathematica

sf[n_] := n!/Quotient[n, 2]!^2; a[n_] := (p = Prime[n]; (sf[p - 1] + (-1)^Floor[(p + 2)/2])/p); Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Jun 28 2013 *)
a[p_] := (Binomial[p-1, (p-1)/2] - (-1)^((p-1)/2)) / p
Join[{1, 1}, a[Prime[Range[3, 20]]]] (* Peter Luschny, May 13 2017 *)

Prog

(PARI) a(n, p=prime(n)) = ((p-1)!/((p-1)\2)!^2 - (-1)^(p\2))/p \\ David A. Corneth, May 13 2017

Crossrefs

Cf. A163213, A002068, A163212, A163209, A007619, A007540.

Sequence in context: A121984 A107177 A096207 * A163211 A126335 A256329

Adjacent sequences: A163207 A163208 A163209 * A163211 A163212 A163213

Keyword

nonn

Author

Peter Luschny, Jul 24 2009

Status

approved