Now we investigate the potential values a, b, c can be

So to recap we know:

1. a = 3k + 2, k is an integer

2. b > 0

3. (a + 1) and (8a - 1) are divisible by 3

4. b is a square divisor of ((a + 1)(a + 1)(8a - 1)) // 27

5. c = ((a + 1)(a + 1)(8a - 1)) / (27b^2)

On last thing I did was to bound k. If not the code takes way too long. 

Now we have what we need to solve the problem.

I outline the algorithm

1. Initialize count = 0

2. Loop through all possible k

   1. a = 3k + 2

   2. t = ((a + 1)(a + 1)(8a - 1)) // 27

   3. Loop through all square divisors, b, of t

      1. c = t / (b^2)

      2. If a + b + c < 110,000,000

         1. count += 1

3. Return count

Step 2.3 is of special importance.

Instead of finding divisors of t directly we can find prime factors of (a + 1)/3 and (8a - 1)/3 and then get the prime factors of t from there. Once we have the prime factors of t, we can recursively generate the square divisors of t.

I include my code for this section below as it can be a little tricky to implement, take the time to read it carefully and you will see that it's not so hard!