I recently read that if the number of multiplicative partitions of $n$ is $a_n$, McMahon and Oppenheim observed that its Dirichlet series generating function $f(s)$ has the product representation
$$f(s)=\sum_{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}=\prod_{k=2}^{\infty }{\frac {1}{1-k^{-s}}}.$$
What is the proof for this? I didn't find any complete proof to the above correspondence. Does someone know where to find it? Or at least how it works?