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Alexandrian Integers

Problem 221

Published on Saturday, 13th December 2008, 01:00 pm; Solved by 1162

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

A = p · q · r    and  
1
A
 =
1
p
 +
1
q
 +
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.