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 $$\dfrac{1}{A} = \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{r}.$$

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 $2^N$th Alexandrian integer.