Complexity of a certain leaf language with Prime & Composite number of accepting paths.
@Tayfun Pay, you are trying to get us to solve a problem for you!  

As it turns out, you only need to apply primality on the count of the number of accepting paths and this count requires polynomial number bits to write down. But the concatenated string on the paths of the NP machine is exponentially long. It is as if the primality algorithm is run on a "padded/unary" version. So it is quite weak and runs within logspace and hence the whole things is within PSPACE.

Now, if you refer to the table (Figure 1) in this paper by [Jenner et al], (or directly to [Hertrampf et all 93]), you will notice that classes within PSPACE (ModP/PH etc) emerge only when the leaf language is a subset of regular languages such as solvable/aperiodic etc. But as is well known, unary PRIMES is not regular (by standard application of pumping lemma).

This much I know. Now to your question. I would suspect PSPACE is the right bound and the proof should not be too difficult, given the techniques from the papers quoted above. If you spend some time thinking, I am sure you will be able to prove the correct bound one way or the other. And especially if these are your first steps into research, I will say, go for it!