Let's say that the linear form $ax+by$ represents $n$ if $ax+by=n$ for some positive integer $x$ and $y$.

Call a pair $(a,b)\in\Bbb N\times\Bbb N$ with $\mathsf{gcd}(a,b)=1$ *excellent* if linear form $ax+by$ has following property: For each composite $n<(a-1)(b-1)$ (the Frobenius number of $(a,b)$) represented by the linear form there is exactly one collection of divisors starting from some $t_j\geq t_1>a,b$ to $t_s\geq t_j$ at every $i\geq1$ those $t_{i}$ with $t_j|t_{i}$ in $a,b<t_1<\dots<t_s\leq n$ is represented by the linear form and no other divisors are represented.

Do excellent pairs exist at all?

If they do, then is it true that for every sufficiently large integer $l$, there is a excellent pair $(a,b)$ with $a,b\in[l,2l]$?

Note that every *excellent* pair is a *good* pair in Problem related to Frobenius coin problem and so *excellent* pair is a stronger condition.

quitehard to parse with the nested conditionals. $\endgroup$ – user9072 Dec 1 '15 at 20:51