I am looking for the number of first relatively large set of consecutive positive integers generated by
$$ \{am+bn:\ 0\leq m\leq x,\ 0\leq n\leq y \} $$ where $a,b$ are given positive integers and $\gcd(a,b)=1$.
For instance; let $a=2$, $b=3$, $x=y=5$, then $$ S=\{2,3,4,\ldots,23\} $$
and let $a=3$, $b=10$, $x=y=20$, then $$ S=\{18,19,20,\ldots,242\} $$ of course the set generated by $am+bn$ with given bounds on $m,n$ is larger but I am looking for the number of the first set of generated successive integers. Is there any know results about this cardinal in terms of $x,y$?
I did some experiment when $x=y$ and I guess this cardinal is about $O(x\log x/\log\log x)$. Am I right? Maybe my question needs to be more clear yet.