Determine the probability that for a positive base ten integer N drawn at random between 2 and 201 inclusively, the number N^{3} - 1 is expressible in the form p*q*r, where p, q and r are three distinct positive integers such that p, q and r (in this order) corresponds to three consecutive terms of an arithmetic progression.

Well, that's not the most general formal for a geometric progression, which means that the earlier solution outline might be missing solutions.

a, a+b, a+2b is general,

which means that N^3 - 1 must be a(a^2 +3ab+2b^2).

It might be easier to work with

q-a, q, q+a

which means that N^3 - 1 must be q(q^2 -a^2) where q > a.