A proper general recursive function which grows slower than a primitive recursive function.

Question

Does there exist a general recursive function which is not primitive recursive, which grows slower than some primitive recursive function? In fact, is there such a function which is bounded by a constant function?

Answer 1

Yes, via direct diagonalization: let $$f(x)=\begin{cases} 1 & \mbox{ if } p_x(x)=0,\\ 0 & \mbox{ if } p_x(x)>0\\ \end{cases}$$ where $(p_n)_{n\in\mathbb{N}}$ is the usual effective enumeration of the primitive recursive functions. This $f$ is computable, not primitive recursive, and as bounded as one could hope.