Square Digit Chains

A number chain is created by continuously adding the square of the
digits in a number to form a new number until it has been seen before.
For example, 44 → 32 → 13 → 10 → 1 → 1 and 85 → 89 → 145 → 42 → 20 → 4
→ 16 → 37 → 58 → 89. Therefore any chain that arrives at 1 or 89 will
become stuck in an endless loop. What is most amazing is that EVERY
starting number will eventually arrive at 1 or 89. How many starting
numbers below ten million will arrive at 89?