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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?

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In "Extensions of some theorems of Gödel and Church" it's shown by Barkley Rosser that these sets are exactly the recursive sets:

Corollary I. If a class can be enumerated (allowing repetitions) by a general recursive function, it can be enumerated (allowing repetitions) by a primitive recursive function.

Note that the crux here is repetitions. Since you constructed sets, and comparing them with other sets, any repetitions do not violate their equality.

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