We introduce the notion of "semi-r.e." for subsets of $\omega$, a generalization of "semirecursive" and of "r.e.", and the notion of "weakly semirecursive", a generalization of "semi-r.e.". We show that $A$ is weakly semirecursive iff, for any $n$ numbers $x_1,\ldots,x_n$, knowing how many of these numbers belong to $A$ is equivalent to knowing which of these numbers belong to $A$. It is shown that there exist weakly semirecursive sets that are neither semi-r.e. nor co-semi-r.e. On the other hand, we exhibit nonzero Turing degrees in which every weakly semirecursive set is semirecursive. We characterize the notion "$A$ is weakly semirecursive and recursive in $K$" in terms of recursive approximations to $A$. We also show that if a finite Boolean combination of r.e. sets is semirecursive then it must be r.e. or co-r.e. Several open questions are raised.