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.
Categories: math.NT