Any r.e. subset of $A\subseteq\mathbb{N}$ which contains the set $$\mathrm{Tot}=\{i\mid i\ \mbox{is an index of a total function } f\}$$ must, by a standard argument (of Post?) contain some partial recursive function indices.
Given a partial function index (and every total function), it's pretty easy to construct many others, e.g. an index for the function which returns $0$ on prime inputs and is undefined otherwise.
But must $A$ contain all partial functions? This seems like a simple question, but I can't find an argument one way or the other.
Edit: I'm equally (more, actually) interested in the case where $A$ is recursive, e.g., represents the programs from some programing language.