A351986 - OEIS

%I #16 Mar 17 2022 23:58:19

%S -4,-2,-1,-1,-1,-1,-1,-2,-1,1,-1,2,-4,2,-1,1,-18,-6,-2,-1,-2,-2,-2,-3,

%T -2,2,-2,3,-18,6,-2,1,2,-2,2,-1,2,2,2,1,-48,-12,-3,-1,-12,-6,-3,-2,-3,

%U -3,-3,-4,-3,3,-3,4,-12,6,-3,2,-48,12,-3,1,12,-6,3,-1,3,-3,3,-2,3,3,3,2,12,6,3,1

%N Four-column table read by rows, giving quadruples of integers [w,x,y,z] such that y^2 - y - x*z = 0 and x^2 = w*y with w*y != 0 and y != 1, sorted by the absolute value of y with the negatives first, then by x in ascending order.

%C When [w,x,y,z] is a row, f(a,b,c) = w*a*b*c + x*(a*b + a*c + b*c) + y*(a+b+c) + z is associative in the following sense. f((a,b,c),d,e) = f(a,f(b,c,d),e) = f(a,b,f(c,d,e)) for all a,b,c,d,e. f(a,b,c) is commutative because of its symmetry.

%C For each quadruple, the corresponding f(a,b,c) has a unique zero element (call it theta), meaning f(a,b,theta) = f(a,theta,b) = f(theta,a,b) = theta for all a,b. Theta = -y/x = - x/w. f(a,b,c) also has not one but two identity elements (id_1 and id_2), meaning f(a,id_1,id_1) = f(id_1,a,id_1) = f(id_1,id_1,a) = a for all a and f(a,id_2,id_2) = f(id_2,a,id_2) = f(id_2,id_2,a) = a for all a. Id = (-y +- sqrt(y))/x = theta +- sqrt(y)/x. Thus theta = (id_1 + id_2)/2.

%C The identity elements and theta are integers when y is a square and x divides sqrt(y).

%H David Lovler, <a href="/A351986/b351986.txt">Table of n, a(n) for n = 1..10792</a>

%H David Lovler, <a href="/A351986/a351986.txt">The first 2698 quadruples for y up to 100</a>.

%e Table begins:

%e   [  w,   x,  y,  z]

%e   -------------------

%e   [ -4,  -2, -1, -1];

%e   [ -1,  -1, -1, -2];

%e   [ -1,   1, -1,  2];

%e   [ -4,   2, -1,  1];

%e   [-18,  -6, -2, -1];

%e   [ -2,  -2, -2, -3];

%e   [ -2,   2, -2,  3];

%e   [-18,   6, -2,  1];

%e   [  2,  -2,  2, -1];

%e   [  2,   2,  2,  1];

%e   [-48, -12, -3, -1];

%e   [-12,  -6, -3, -2];

%e   [ -3,  -3, -3, -4];

%e   [ -3,   3, -3,  4];

%e   [-12,   6, -3,  2];

%e   [-48,  12, -3,  1];

%e   [ 12,  -6,  3, -1];

%e   [  3,  -3,  3, -2];

%e   [  3,   3,  3,  2];

%e   [ 12,   6,  3,  1];

%e   ...

%o (PARI) { my(y=1); fordiv (y^2+y, x, print([-((y^2+y)/x)^2/y, -(y^2+y)/x, -y, -x]) );

%o fordiv (y^2+y, x, print([-(x^2/y), x, -y, (y^2+y)/x]) );

%o for (y = 2, 6, fordiv (y^2+y, x, if(type(w = -(((y^2+y)/x)^2)/y)=="t_INT", print([w, -(y^2+y)/x, -y, -x]) ));

%o fordiv (y^2+y, x, if(type(w = -x^2/y)=="t_INT", print([w, x, -y, (y^2+y)/x]) ));

%o fordiv (y^2-y, x, if(type(w = (((y^2-y)/x)^2)/y)=="t_INT", print([w, -(y^2-y)/x, y, -x]) ));

%o fordiv (y^2-y, x, if(type(w = x^2/y)=="t_INT", print([w, x, y, (y^2-y)/x]) )) )}

%Y Cf. A332083.

%Y The rows of A351581 are a subset.

%K sign,tabf

%O 1,1

%A _David Lovler_, Feb 27 2022