As an example of why I don't think this is as easy as everyone seems to think:

shown for 1-20

Probability of a= any given number, 1/20.
Probablility of b = any given sum of a specific set of factors, 1/20

So each number 1-20 = 1/20 of being picked, then (1/20)*# of potential b's for the a.

(colours and underline are for aesthetic purposes only, intending to improve readibility)

# of potential b's can be found through prime factorization:

# -> Factors -> potential b's -> probability for b's
1 -> 1,1 -> 2 -> 1/20
2 -> 1,2 -> 3 -> 1/20
3 -> 1,3 -> 4 -> 1/20
4 -> 1,2,2 -> 5,4 -> 2/20
5 -> 1,5 -> 6 -> 1/20
6 -> 1,2,3 -> 7, 5 -> 2/20
7 -> 1,7 -> 8 -> 1/20
8 -> 1,2,2,2 -> 9, 6 -> 2/20
9 -> 1,3,3 -> 10, 6 -> 2/20
10 -> 1,2,5 -> 11, 7 -> 2/20
11 -> 1,11 -> 12 -> 1/20
12 -> 1,2,2,3 -> 13, 8, 7 -> 3/20
13 -> 1,13 -> 14 -> 1/20
14 -> 1,2,7 -> 15,9 -> 2/20
15 -> 1,3,5 -> 16,8 -> 2/20
16 -> 1,2,2,2,2 -> 17, 10, 8 -> 3/20
17 -> 1,17 -> 18 -> 1/20
18 -> 1,2,3,3 -> 19, 11, 9 -> 3/20
19 -> 1,19 -> 20 -> 1/20
20 -> 1,2,2,5 -> 12, 9 -> 2/20

so the answer is 1/20*(34/20) = .085

So for 1-20, there is an 8.5% chance of choosing an A, B pair where the product equals A and the sum equals B.

(unless I made a mistake, but still this shows the complexity, I think)

So I don't see how you can come up with a simple equation that will be able to take into account that every number will have a different number of potential B's. It seems to me that it would need to be able to find how many factors each number had, including repeats, and how many numbers they could sum to, excluding repeats. And this for every number through 2007.

(unless there is a pattern, but I doubt there is, because if there was, we would be able to use it to calculate prime numbers, to my knowledge, despite many efforts by many great mathematicians, this has not happened yet.)

If you can solve the problem for the first 20, like I have, without using tedious brute techniques, like I did, please post, I would be very interested to see it done. (If you get close, you might want to check my work, make sure I didn't miss something, and am giving you a false answer to shoot for)

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