I'm currently working in the following exercise:
Let $T$ be the function that sum the digits of an integer $n$. Let $a (n)$ be the number of times we must apply $T$ to an integer n so that it becomes a fixed point. For example $a(452) = 2$, since $T (452) = 11$, $T (11) = 2$, $T (2) = 2$. So we must apply twice T to get to the fixed point $2$. Find the smallest positive integer $n$ such that $a (n) = 3$. Find the smallest positive integer n such that $a (n) = 4$.
I've been trying descomposition in prime factors so that could help me to find the numbers required to find the answer but that option is not working. If anyone has a hint or any help will be really appreciated.