OFFSET
1,1
COMMENTS
It is the same to sort by y then by w also to sort by y then by z descending.
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 are integers when y is a square and x divides sqrt(y).
LINKS
David Lovler, Table of n, a(n) for n = 1..2720
David Lovler, The first 680 quadruples for y up to 100.
FORMULA
Looking at A336013, if [X,Y,Z] is a row and f(a,b) = X*a*b + Y*(a+b) + Z is the corresponding associative function with id = -Z/Y and theta = -Y/X, then the composition f(f(a,b),c) = w*a*b*c + x*(a*b + a*c + b*c) + y*(a+b+c) + z = f(a,b,c) gives the quadruple [w,x,y,z]. f(a,b,c) has the same theta as f(a,b); the two identity elements for f(a,b,c) are id and 2*theta - id.
If theta and the identity elements are computed from a quadruple, f(a,b,c) can be written as (a*b*c - theta*(a*b + a*c + b*c) + theta^2*(a+b+c) - theta^3)/(id-theta)^2 + theta. The square in the denominator ensures that f(a,b,c) is the same for either id.
Two parameters are sufficient to describe a row. For n*s > 1, rows are [w,x,y,z] = [n, n*s, n*s^2, (n^2*s^4-n*s^2)/(n*s)] = [n, n*s, n*s^2, n*s^3 - s]. In terms of n and s, theta = -s and id = s*(-1 +- 1/sqrt(n)). Rows with s=1 stand out as having w=x=y; theta = -1 and id = -1 +- 1/sqrt(w).
EXAMPLE
Table begins:
[ w, x, y, z]
-----------------
[ 2, 2, 2, 1];
[ 3, 3, 3, 2];
[ 12, 6, 3, 1];
[ 1, 2, 4, 6];
[ 4, 4, 4, 3];
[ 9, 6, 4, 2];
[ 36, 12, 4, 1];
[ 5, 5, 5, 4];
[ 20, 10, 5, 2];
[ 80, 20, 5, 1];
[ 6, 6, 6, 5];
[150, 30, 6, 1];
[ 7, 7, 7, 6];
[ 28, 14, 7, 3];
[ 63, 21, 7, 2];
[252, 42, 7, 1];
[ 2, 4, 8, 14];
[ 8, 8, 8, 7];
[ 98, 28, 8, 2];
[392, 56, 8, 1];
[ 1, 3, 9, 24];
[ 4, 6, 9, 12];
[ 9, 9, 9, 8];
[ 16, 12, 9, 6];
[ 36, 18, 9, 4];
[ 64, 24, 9, 3];
[144, 36, 9, 2];
[576, 72, 9, 1];
...
For row [1, 2, 4, 6], f(a,b,c) = a*b*c + 2*(a*b + a*c + b*c) + 4*(a+b+c) + 6. Theta = -2; id_1 = -1, id_2 = -3. The associative function f(a,b) = a*b + 2*(a+b) + 2 has theta = -2 and id = -1; f(f(a,b),c) = f(a,b,c). Another associative function g(a,b) = -a*b - 2*(a+b) - 6 with theta = -2 and id = -3 likewise gives g(g(a,b),c) = f(a,b,c).
PROG
(PARI) { my(y); for (y = 2, 9, fordiv (y^2-y, x, if(type(w = x^2/y) == "t_INT", print([w, x, y, (y^2-y)/x]) )) ) }
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
David Lovler, Feb 13 2022
STATUS
approved