Let $p$ be a fixed prime and $r$ a fixed prime power. Let $x_{p'}$ denote the largest divisor of a positive integer $x$ such that $x_{p'}$ is not divisible by $p$. (For example, $60_{2'}=15$.) I would like to prove:

**Claim**: $\dfrac{(r^n-1)_{p'}}{n}\rightarrow\infty$ as $n\rightarrow \infty$.

Sorry if this is obvious and thank you for your help.