More precisely, does there exist an unbounded sequence $a_0, a_1, ... \in \mathbb{N}$ of primes such that the function

$\displaystyle O(z) = \sum_{n \ge 0} a_n z^n$

is meromorphic on $\mathbb{C}$?

[A previous version of the question also asked about the exponential generating function of $(a_n)$. However, such a function can trivially be entire. - GJK]