 Solution

Fix an effective numbering of Turing machines and use the diagonal halting setIt is computably enumerable: dovetail all the computations and output each index when its computation halts. It is not a recursive set. Otherwise a machine could, on input $e$, loop if the alleged decider says $e\in H$ and halt otherwise. Applying this machine to its own index contradicts either answer. Hence $H$ is computably enumerable but not a recursive set.