Determine the probability that for a positive base ten integer N drawn at random between 2 and 201 inclusively, the number N³ - 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.

(In reply to Infinite family of solutions by Brian Smith)

I ran the numbers for n=2 to 3000.  There were only 11 solutions with p>2:

100     27  143  259
121     45  152  259
121     28  185  342
196     117  211  305
273     38  527  1016
289     182  307  432
513     56  1112  2168
1024    331  1359  2387
1353    686  1526  2366
2601    350  5102  9854
2745    602  4298  7994

Posted by Brian Smith on 2010-06-04 19:58:53