# What is known about the sets enumerated by primitive recursive functions?

Let's say that a set of natural numbers $$S \subseteq \mathbb{N}$$ is primitive recursively enumerable if there exists some primitive recursive function $$f$$ such that $$S$$ is the range of $$f$$. That is, we can enumerate $$S$$ by calculating $$\{f(0), f(1) \ldots \}$$.

What is known about this class of sets? Where does it stand in terms of computability? I suspect that it contains sets that are not context free, and that it does not contain all recursive sets.

Has this been studied? Does anyone have a reference for this?