  In this paper, the following results are proved: (i) For any odd integer
$\ell$ with at most two distinct prime factors and any positive integer $n$,
the product $(1^\ell+1)(2^\ell+1)\cdots (n^\ell +1)$ is not a powerful number;
(ii) For any integer $r\ge 1$, there exists a positive integer $T_r$ such that,
if $\ell$ is a positive odd integer with at most $r$ distinct prime factors and
$n$ is an integer with $n\ge T_r$, then $(1^\ell+1)(2^\ell+1)\cdots (n^\ell
+1)$ is not a powerful number.
