I came up with the following question. I could not find a way to solve it, and haven't seen it anywhere also. Is this an open problem?
Problem statement: Consider a prime number p. Consider all numbers having it as a factor. Repeatedly sum the digits of those numbers until you get a number less than or equal to p. In this sequence, will you ever get a p? Is it guaranteed that you get for various values of p (larger ones)?
For example, a "sum" is defined as applying + together on all digits and get the resulting number. You apply "sum" again on it, if it is larger than p, of course.