= Solution
Fix an effective numbering of <Turing machines> and use the <diagonal halting set>
$$
H=\{e:M_e(e)\text{ halts}\}.
$$
It 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 \b[$H$ is <computably enumerable> but not a <recursive set>].
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