We investigate a class of generalized calculations on Turing machines with oracles which were considered in papers by \textit{N. V. Belyakin} [Mat. Sb., Nov. Ser. 101(143), 21-43 (1976; Zbl 0342.02031)] and \textit{V. A. Ganov} [''The selector function and the continuum hypothesis'' (Russian) (manuscript deposited at VINITI (Moscow) No.94-81)]. We use as oracles one-place functions not everywhere defined which satisfy the special conditions of regularity, weak groundedness and normalizability. In Section 1 we show that the idea of enumerability of sets with oracles can be substantially strengthened. In Section 2 we introduce the condition of decomposability of an oracle and prove its equivalence to the condition of normalizability.