Pick any whole number greater than 1.

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?

Think about this problem this way. The smallest 21 step number is 924. This means that the smallest 22 step number, must be at least 924 after the first step. That is, the digits of its distinct factors must add to at least 924. Furthermore, the smallest 23 step number, must have factors whose digits add up to at least that number. That's why the values begin to increase so rapidly.

Posted by Justin on 2006-01-13 11:15:24