# An integer-valued function

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• June 24th 2009, 01:24 AM
NonCommAlg
An integer-valued function
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