1. Write down all of its proper divisors (including 1 and the number itself)
2. Add the digits of these divisors.
3. Use this sum to repeat steps 1 and 2 until your number does not change.

Must the process terminate?
At what number(s) can the process terminate at?
What numbers <1000 take the most steps to terminate?

Are you suggesting there's a way other than just pointing the computer in the right direction and letting it run?

I ran numbers up to 10000, I think, and the longest chain was still 21.