Suppose that $A = \{a_n\}$ is an increasing sequence of positive integers and let $\zeta_A(s)=\sum_n \frac{1}{{a_n}^s}$.
For which sequences does $\zeta_A (s)=\prod_{p_{n_k}}(1-{p_{n_k}}^{-s})^{-1}$ for some infinite subset of the primes $\{p_{n_k}\}$?