OMCC 2017 Problem 3

Tita the Frog sits on the number line. She is initially on the integer number $k>1$ . If she is sitting on the number $n$ , she hops to the number $f(n)+g(n)$ , where $f(n)$ and $g(n)$ are, respectively, the biggest and smallest positive prime numbers that divide $n$ . Find all values of $k$ such that Tita can hop to infinitely many distinct integers.