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A163210
Swinging Wilson quotients ((p-1)$ +(-1)^floor((p+2)/2))/p, p prime. Here '$' denotes the swinging factorial function (A056040).
7
1, 1, 1, 3, 23, 71, 757, 2559, 30671, 1383331, 5003791, 245273927, 3362110459, 12517624987, 175179377183, 9356953451851, 509614686432899, 1938763632210843, 107752663194272623
OFFSET
1,4
LINKS
M. E. Bassett, S. Majid, Finite noncommutative geometries related to F_p[x], arXiv:1603.00426 [math.QA], 2016.
Peter Luschny, Swinging Primes.
EXAMPLE
The 5th prime is 11, (11-1)$ = 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
KEYWORD
nonn
AUTHOR
Peter Luschny, Jul 24 2009
STATUS
approved