I have thought that Gödel introduced the concept of Primitive recursive functions in his seminal paper "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme" (I hope I got the title right...). This paper, to best of my knowledge, was published in 1931.

Ackermann published the famous Ackermann function in his paper "Zum Hilbertschen Aufbau der reellen Zahlen" in (again, to best of my knowledge) 1928. His publication seems to enclose primitive recursive functions and the main results is that there exists a total computable function outside PR. This suggests that PR has been known prior 1928, thus prior Gödel's work.

Who introduced primitive recursive functions first?