Given the values of integers $1<a_1<a_2<\cdots <a_n$, consider the linear combination
$q_1{a}_1+q_2{a}_2+\cdots +q_n{a}_n=b$, using only integer values $q_k\ge 0$.

Note that for a given set of $a_k$, it may be that not all values of $b$ are possible.
For instance, if $a_1=5$ and $a_2=7$, there are no $q_1\ge 0$ and $q_2\ge 0$ such that $b$ could be
$1,2,3,4,6,8,9,11,13,16,18$ or $23$.
In fact, $23$ is the largest impossible value of $b$ for $a_1=5$ and $a_2=7$.
We therefore call $f(5,7)=23$.
Similarly, it can be shown that $f(6,10,15)=29$ and $f(14,22,77)=195$.

Find ${\displaystyle \sum f(pq,pr,qr)}$, where $p$, $q$ and $r$ are prime numbers and $p<q<r<5000$.