I am currently studying a combinatorics question that makes appear the following type of sets:
$p\mathbb{Z}\cap [n]+q\mathbb{Z}\cap[n]$.
It is basically interescting ideals of $\mathbb{Z}$, but allowing only to take Bezout combination within $[n]$. If $gcd(p,q)$ can be constructed this way, then it is equal to $gcd(p,q)\mathbb{Z}$ as in the normal case. Otherwise, it has several generators, but I am still very confused on how the configuration looks like. Is there anything already known about that? A typical question (that might be ambitious): given $n$, how many different sets can be constructed, by adding summing several such $p_i\mathbb{Z}\cap [n]$ ?