# Complete $\mu$-recursive function that is not primitive recursive

Could you find an example of a complete $\mu$-recursive function that is not a primitive function?

The definition of primitive recursive functions (whichever one you use) allows enumerating them $f_1,f_2,\ldots$. Since each primitive recursive function is total, the function $f(n) = f_n(n)+1$ is recursive. It is clearly not primitive recursive.