As I read the question, you want find certain pairs (a,b) to use the linear form (ax+by) to represent positively only certain numbers, and only in a certain way. In particular, for any n greater than a and b and less than g(a,b) =ab-a-b, it is either not representable as ax+by for integers x and y where at most one of x and y is 0 and otherwise positive, or else n is so representable and also only certain divisors (which are larger than both a and b) of n are also representable, including at most one maximal divisor of n.
Well, you aren't going to find many such a,b. Let's pick a small representable number s greater than both a and b. To be definite, pick s=a+b. Then 2s, 3s, and 6s are also representable, so by your condition 7(a+b) is greater than ab. So the smaller of a and b is less than 14, and as b gets large a is bounded above by 7. And we haven't explored what happens when you pick ceiling(b/a)*a. I predict there will be only finitely many such pairs.
Gerhard "Leaving The Rest To You" Paseman, 2015.12.01