I wish there were more, but the subject pretty much captures my whole question.
Is there a non-Turing-complete model (some constrained term rewriting system or automaton or what have you) which is known to be able to enumerate the prime numbers, all of the prime numbers (well, til you pull the plug on the algorithm), and only the prime numbers?
To be clear, one of the criteria I'm imposing is that this algorithm would be non-halting, and so long as it is left running, it will continue intermittently outputting prime numbers. This seems to necessitate an unbounded working memory, which gets you a lot of the way towards Turing-complete already.
Furthermore, this must be a model with a finite description, meaning no cute "consider the system mapping the natural numbers to the primes" answers, please. Even if technically correct, what I'm really after is whether it seems probable that prime enumeration would be a strong indicator of Turing completeness.
Edit: As for the objection that Turing machines can't yield values at will, I consider that semantics and not relevant to the spirit of the question. A Turing machine could certainly record all prime numbers found thus far on its tape, which we could presumably examine at will.