SCHWEITZER 1982 Problem 4

Let $$f(n)= \sum_{p|n , p^{\alpha} \leq n < p^{\alpha+1} \ } p^{\alpha} .$$ Prove that $$\limsup_{n \rightarrow \infty}f(n) \frac{ \log \log n}{n \log n}=1 .$$

P. Erdos