Let define the Exactly $A$-SAT problem : Given a CNF formula $F$ and a set $A$ of positive integers, is it true that the number of models of $F$ is in $A$?What is the complexity of Exactly $A$-SAT  ?Edit -- Some different cases seem interesting:Let $A_{MAJ}=\{m\in\mathbb{N},2^{n-1} < m\leq2^{n}\}$, then Exactly $A_{MAJ}$-SAT corresponds to the canonical $PP$-complete MAJ-SAT (Given a CNF formula $F$, is it true that $F$ is satisfied by more than half of the possible variable assignments?) - as long as $A_{MAJ}$ is not considered as part of the input, otherwise, as @Ryan's comment underlines it, the problem is in $P$.Let $A_{SAT}=\{m\in\mathbb{N},1 \leq m\leq2^{n}\}$, then Exactly $A_{SAT}$-SAT is simply SAT.($n$ is the number of variables of $F$).Thank you for your answers and your comments. 
Complexity of Exactly $A$-SAT
cc.complexity theory;sat;counting complexity;polynomial hierarchy