Define f: \mathbb{N} \longrightarrow \mathbb{N} by f(1)=1 and f(n)=\sum_{j=1}^k p_j^{r_j}, where \prod_{j=1}^k p_j^{r_j} is the prime factorization of n. Prove that for any positive integers x_1, \cdots , x_m we have f(\text{lcm}\{x_1, \cdots , x_m\}) \leq \sum_{j=1}^m x_j.

Source: American Mathematical Monthly