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Problem 221

Alexandrian Integers

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

$$A=p·q·r \text{ and } \frac{1}{A}=\frac{1}{p}+\frac{1}{q}+\frac{1}{r}$$

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.

$$A=p·q·r \text{ 且 } \frac{1}{A}=\frac{1}{p}+\frac{1}{q}+\frac{1}{r}$$