A351581 -id:A351581 - cp's OEIS frontend

A351986 Four-column table read by rows, giving quadruples of integers [w,x,y,z] such that y^2 - y - xz = 0 and x^2 = wy with w*y != 0 and y != 1, sorted by the absolute value of y with the negatives first, then by x in ascending order.

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

Offset: 1

David Lovler, Feb 27 2022

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

			Table begins:
  [  w,   x,  y,  z]
  -------------------
  [ -4,  -2, -1, -1];
  [ -1,  -1, -1, -2];
  [ -1,   1, -1,  2];
  [ -4,   2, -1,  1];
  [-18,  -6, -2, -1];
  [ -2,  -2, -2, -3];
  [ -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,  -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];
  ...

David Lovler, Table of n, a(n) for n = 1..10792
David Lovler, The first 2698 quadruples for y up to 100.

Cf. A332083.

The rows of A351581 are a subset.

{ my(y=1); fordiv (y^2+y, x, print([-((y^2+y)/x)^2/y, -(y^2+y)/x, -y, -x]) );
fordiv (y^2+y, x, print([-(x^2/y), x, -y, (y^2+y)/x]) );
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]) ));
fordiv (y^2+y, x, if(type(w = -x^2/y)=="t_INT", print([w, x, -y, (y^2+y)/x]) ));
fordiv (y^2-y, x, if(type(w = (((y^2-y)/x)^2)/y)=="t_INT", print([w, -(y^2-y)/x, y, -x]) ));
fordiv (y^2-y, x, if(type(w = x^2/y)=="t_INT", print([w, x, y, (y^2-y)/x]) )) )}

cp's OEIS Frontend

A351986 Four-column table read by rows, giving quadruples of integers [w,x,y,z] such that y^2 - y - xz = 0 and x^2 = wy with w*y != 0 and y != 1, sorted by the absolute value of y with the negatives first, then by x in ascending order.

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cp's OEIS Frontend

A351986 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A351986 Four-column table read by rows, giving quadruples of integers [w,x,y,z] such that y^2 - y - xz = 0 and x^2 = wy with w*y != 0 and y != 1, sorted by the absolute value of y with the negatives first, then by x in ascending order.