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Alexandrian Integers

 Published on Saturday, 13th December 2008, 01:00 pm; Solved by 1926;
Difficulty rating: 65%

Problem 221

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 6th Alexandrian integer, the first 6 Alexandrian integers being: 6, 42, 120, 156, 420, and 630.

Find the 150000th Alexandrian integer.