Definition. A prime number $p$ is called strange if there exists $k>1$ such that each prime divisior of $p^k-1$ divides $p-1$.
Example 3. The prime number $p=3$ is strange as $3^2-1=8$ has the same prime divisiors as $3-1=2$.
Example 5. The prime number $p=5$ is not strange, since for every $k>1$ the number $5^k-1$ is not a power of $2$ (by the Mihailescu Theorem). By the same reason the prime number $p=17$ is not strange.
Example 7. The prime number $p=7$ is strange since $7^2-1=48$ has the same prime divisors as $7-1=6$.
Example 31. The prime number $p=31$ is strange because $31^2-1=2^6\times 3\times 5$ has the prime divisors as $31-1=2\times 3\times 5$.
Using the small Fermat Theorem, it is possible to prove the following characterization
Theorem. A prime number $p$ is not strange if and only if for every prime divisor $q$ of $p-1$ the number $p^q-1$ has a prime divisor that does not divide $p-1$.
This theorem implies that the prime numbers $11,13,19,23,29,37,41,43,47,53,61,67,71,73,79,83,89,97$ are not strange.
Therefore, among prime numbers $<100$ only 3,7, 31 are strange. All these numbers are Mersenne numbers. In his comment Yaakov Baruch observed that each Mersenne number is strange. So, we can ask
Question 1. Is each strange prime number Mersenne prime?
Question 2. Is the set of non-strange numbers infinite?
Question 3. Is it true that for any number $x$ and prime numbers $p_1,\dots,p_n$ that not divide $x$, the arithmetic progression $x+p_1\dots p_n\mathbb Z$ contains a non-strange prime number?