User blog comment:Pellucidar12/Peng-Pellucid-Armstrong Number/@comment-29915175-20170102011204/@comment-5529393-20170103000147

Hmm, I don't think the function is recursive. It will be recursive if there must be a ZFC proof that a particular integer k is the unique integer satisfying a formula \(\phi\); but if we only require that ZFC prove that there exists a unique integer satisfying \(\phi\), then one can write formulas for Busy Beaver numbers, for instance, and so this function will outgrow BB(n).