I may have the notation wrong, or in a nonstandard way; so here I am assuming $a$, $p$, and $n$ are positive integers, $p$ is prime, and $\gcd(a, p)=1$.
Then by $\mathrm{ord}(a, p^n)$ I mean the order of $a$ in the group $\mathbb Z/p^n\mathbb Z^\times$, which is the least positive exponent $k$ where $a^k\equiv 1$, modulo $p^n$.
Based on really simple examples I had noticed that in general $\mathrm{ord}(a, p)$ may not be $p-1$, even if $a$ is prime. However, in the examples I tried, if $a$ is prime, then $\mathrm{ord}(a, p^{n+1})=p^n \cdot \mathrm{ord}(a, p)$.
Is this actually true in general?
NB: This requires the condition that $a$ is prime. As remarked in the comments, this is not true if $a$ is not prime (although there are weaker theorems that can be proven).
NB2: This is not a question about primitive roots. We do not assume $a$ is a primitive root mod $p^n$.
