Eccoci.

We shall call a positive integer $A$ an "Alexandrian integer", if there exist integers $p, q, r$ such that:

$$A=p\cdot q\cdot r$$ and $${\displaystyle \frac{1}{A}}={\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}+{\displaystyle \frac{1}{r}}.$$

For example, $630$ is an Alexandrian integer ($p = 5, q = -7, r = -18$). In fact, $630$ is the $6$^th Alexandrian integer, the first $6$ Alexandrian integers being: $6, 42, 120, 156, 420$, and $630$.

Find the $150000$^th Alexandrian integer.