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.  
Division by two of functions in #P
cc.complexity theory;complexity classes;counting complexity