Existence of primes $p$ such that all the prime divisors of $p+1$ divide $p-1$

Question

This question recently came up to me in a project and is not taken from a textbook. I would like to know if any characterization of such primes is known from literature. They are seemingly rare but do exist. The first five examples are: $3$, $7$, $31$, $127$, $8191$. Note that these are all congruent to $3$ modulo $4$. The first four are also exponents of Mersenne primes, although $8191$ is not. I have also noted that they all satisfy $p^2 \equiv -1 \pmod{\operatorname{rad}((p+1)(p^2+1))}$, where $\operatorname{rad}(k)$ denotes the product of the prime divisors of an integer $k$. In particular this would imply that $\operatorname{rad}(p+1) \mid \operatorname{rad}(p^2+1)$ as well, and that $-1$ is a quadratic residue modulo each of the prime divisors of $(p + 1)(p^2 + 1) = (p^4-1)/(p-1)$; hence each prime divisor of $(p+1)$ must be congruent to $1$ modulo $4$. Many thanks for any insights.