The existence of such numberings was established by
Richard M. Friedberg in 1958 (Cutland 1980:78).
References
Nigel Cutland (1980), Computability: An Introduction to Recursive Function Theory, Cambridge University Press.
ISBN9780521294652.
Richard M. Friedberg (1958), Three Theorems on Recursive Enumeration. I. Decomposition. II. Maximal Set. III. Enumeration Without Duplication, Journal of Symbolic Logic 23:3, pp. 309–316.
Nikolaj K. Vereščagin and A. Shen (2003), Computable Functions, American Mathematical Soc.