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 $n$ with $a,b<n<g(a,b)=ab-a-b$, it is either not representable as $ax+by$ for integers $x$ and $y$ where both $x$ and $y$ is at least $1$.
Well, you aren't going to find many such $a,b$. Let's pick $s=a+b$. Then $2s, 3s, 6s$ are also representable, so by your condition $7(a+b)>ab$. 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
