KURSCHAK 1985 Problem 2

For every $n\in\mathbb{N}$ , define the power sum of $n$ as follows. For every prime divisor $p$ of $n$ , consider the largest positive integer $k$ for which $p^k\le n$ , and sum up all the $p^k$ 's. (For instance, the power sum of $100$ is $2^6+5^2=89$ .) Prove that the power sum of $n$ is larger than $n$ for infinitely many positive integers $n$ .