In a sense the Ackermann function is not primitive recursive (PR) because it grows too fast.

Are there total recursive, not PR, small functions?

Using a diagonal argument, we may define a total recursive, not PR, and small (the codomain is {0,1}) function as: $f(n)=0$ if $\phi_n(n)\neq 0$, $f(n)=1$ if $\phi_n(n)=0$ where $\phi_i$ is the $i$th PR function. But, to me, this is not a "natural" function and furthermore it depends on the particular $\phi_i$ used.

And the question is: are there total recursive, not PR, natural, and small functions?

To be specific, let "small" mean "takes only the values 0 and 1", and let "natural" mean "recursively defined" (like the Ackermann function).

I apologize if this question is not appropriate to this forum.

Armando