Comments on A000926 Date: Wed, 05 Dec 2007 04:59:24 -0500 From: Eric Rains (rains(AT)caltech.edu) The condition (6) n is not of the form ab+ac+bc with 0 < a < b < c. is equivalent to (5) The class group C(-4n) is isomorphic to (Z/2Z)^m for some integer m. Proof: The basic idea is that if [a b] [b c] is a reduced (i.e. 2|b|<=a<=c) binary form of determinant n with b>=0, then one has 0<=b<=a-b<=c-b and b(a-b)+b(c-b)+(a-b)(c-b)=n, and vice versa. In other words, solutions to de+ef+fd=n with 0<=d<=e<=f are in one-to-one correspondence with reduced binary forms of determinant n with b>=0. The excluded cases d=0,d=e, or e=f correspond to the cases b=0, a=2b or a=c for binary forms. It follows that the solutions with 0<d<e<f are precisely the forms which are not of order two in the class group. In other words, the sequence of numbers n not of the form ab+ac+bc with 0 < a < b < c is the same as the sequence of numbers n such that the class group of \Z[\sqrt{-n}] is of exponent 1 or 2.