$\text{Mod}_k\text{P}$ is the class of decision problems solvable by an NP machine such that the number of accepting paths is divisible by k, if and only if the answer is no.For constant $k$, one can find the number of accepting paths of an NP machine mod $k$ by querying a $\text{Mod}_k\text{P}$ oracle with $0$,$1$,$2$,$\ldots$,$k-1$ accepting paths added to the original NP machine. Are there more interesting functions one can compute with oracle access to a $\text{Mod}_k\text{P}$-complete problem?  For instance, does $\text{P}^{\text{Mod}_k\text{P}} =\text{P}^{\#\text{P}}$? It is clear that $\text{P}^{\text{Mod}_k\text{P}} \subseteq \text{P}^{\#\text{P}}$.For $k=2$, we have that $\text{Mod}_k\text{P} = \oplus \text{P}$, and certain hard problems (like computing the permanent) become easy mod 2.  Therefore, we might expect the power of $\text{P}^{\text{Mod}_k\text{P}}$ to depend on $k$.  
$\text{P}^{\text{Mod}_k\text{P}}$ vs $\text{P}^{\#\text{P}}$?
cc.complexity theory