%I A080908
%S A080908 0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,0,0,1,0,1,1
%V A080908 0,1,-1,1,-1,0,0,1,0,-1,1,0,0,1,0,1,0,0,1,0,0,0,1,0,1,-1
%N A080908 a(n) = the sign of r(n), where r(n) is the integer in [ -2n,2n] which 
               is congruent to (2n)! modulo 4n+1
%C A080908 If 4n+1 is composite, then a(n)=0, except when n=2. If 4n+1 is a prime 
               number, then (2n)! is a square root of -1 modulo 4n+1 and a(n)=1 
               or a(n)=-1. Is there a simple way to predict whether a(n)=1 or a(n)=-1 
               ? The Maple program could be simplified by setting sign(0)=0, but 
               I do not know how to do that.
%D A080908 Hardy, G. H. and Wright, E. M., An introduction to the theory of number 
               (Fourth Edition, 1960), section 7.7: the residue of ((p-1)/2)!
%e A080908 a(2) = -1 because 4! = 24 = -3 modulo 9 and a(5) = 0 because 10! = 0 
               modulo 21.
%p A080908 for n from 0 to 100 do (sign(2*mods((2*n)!,4*n+1)+1) + sign(2*mods((2*n)!,
               4*n+1)-1))/2 end do;
%Y A080908 Sequence in context: A138886 A099859 A102460 this_sequence A131720 A131719 
               A100656
%Y A080908 Adjacent sequences: A080905 A080906 A080907 this_sequence A080909 A080910 
               A080911
%K A080908 sign
%O A080908 0,1
%A A080908 Christophe Leuridan (Christophe.Leuridan(AT)ujf-grenoble.fr), Apr 01 
               2003