Products of shifted primes simultaneously taking perfect power values

Let $r\ge 2$ be an integer and let $A$ be a finite, nonempty set of nonzero integers. We will obtain a lower bound for the number of squarefree integers $n$, up to $x$, for which the products ${\prod }_{p\mid n}(p+a)$ (over primes $p$) are perfect $r$th powers for all $a\in A$. Also, in the cases $A=\{-1\}$ and $A=\{+1\}$, we will obtain a lower bound for the number of such $n$ with exactly $r$ distinct prime factors.