Three positive integers are chosen at random without replacement from 1,2,....,64. What is the probability that the numbers chosen are in geometric sequence?

Order of choice doesn't matter. For example, 4-1-2 would qualify as numbers in geometric sequence.

Bonus Question:
Generalise this result (in terms of n) covering the situation where three positive integers are chosen at random without replacement from 1,2,.....,n².

I now count 58 geometric sequences.  Each of the following is a possible first term followed the ratio of subsequent terms. 
Square-free numbers must have integer ratios but numbers that are not square-free can have non-integer ratios and still have integer terms.

1 ratios 2, 3, 4, 5, 6, 7, 8
2 ratios 2, 3, 4, 5
3 ratios 2, 3, 4
4 ratios 3/2, 2, 5/2, 3, 7/2, 4
5 ratios 2, 3
6 ratios 2, 3
7 ratios 2, 3
8 ratios 3/2, 2, 5/2
9 ratios 4/3, 5/3, 2, 7/3, 8/3
10 ratio 2
11 ratio 2
12 ratios 3/2, 2
13 ratio 2
14 ratio 2
15 ratio 2
16 ratios 5/4, 6/4, 7/4, 2
18 ratios 4/3, 5/3
20 ratio 3/2
24 ratio 3/2
25 ratios 6/5, 7/5, 8/5
27 ratio 4/3
28 ratio 3/2
32 ratio 5/4
36 ratios 7/6, 8/6
49 ratio 8/7

This makes a general count leading to the bonus much more complicated than before.

Edited on February 28, 2013, 1:42 pm

Posted by Jer on 2013-02-28 11:30:27