In an explicit construction in combinatorics I need to study the following problem: assume we pick a odd prime number $p$, a generator $g$ of the multiplicative group $(Z/pZ)^{\ast}$.

Question 1: for how many $x\in\{2,\dots,p-2\}$, is it true that $x\equiv g^x\, (p)$.

Question 2: how the previous counting depends on the choice of $g$?

In general I would need indeed the number of solutions of the congruence equation $x\equiv a+g^x\, (p)$ (for fixed $a$), but already the base case $a=0$ is interesting for me.