Division by two of functions in #P Let $F$ be an integer valued function such that $2F$ is in $\#P$.  Does it follow that $F$ is in $\#P$?  Are there reasons to believe this is unlikely to always hold?  Any references I should know about?

Somewhat surprisingly, this situation came up (with a much larger constant), for a function $F$ for which $F \in? \#P$ is an old open problem.  

Note: I am aware of the paper M. Ogiwara, L. Hemachandra, A complexity theory for feasible closure properties where a related division-by-2 problem has been studied (see Thm 3.13).  Their problem is different however, as they define the division for all functions via the floor operator.  That allowed them to make some quick reductions to parity problems.
I try to give my intuition why I think this is unlikely to hold. Take your favorite problem in $PPA$, and convert it into a problem in $\sharp P$, e.g., our function $f$ can be the number of Hamiltonian cycles in an input 3-regular graph containing a certain fixed edge. From the parity argument we know that $f$ is always even, so you can define $F:=f/2$ and I see no reason why $F$ would be in $\sharp P$.