# Can a subset of indexes of a partial recursive function be recursively enumerable but not recursive?

Consider a Gödelian numbering of partial recursive functions. Consider the set of indexes corresponding to some function $$\phi$$. It may be seen that this set is not recursively enumerable. Does this set contain a recursively enumerable non-recursive subset?

I guess the answer is yes, but please check critically. One can easily construct an infinite recursive set of indices of $$\phi$$ (adding "comments" to a program). From this infinite recursive set one can pick out a non recursively enumerable subset.