Number of integers for which $np(n)$ is a perfect square

As suggested by Gerry Myerson, I examined OEIS but made a computational error. Thanks @wojowu

According to Magma, there are no values of $n \in (3,10000]$ for which $np(n)$ is a perfect square.

I've tested if $np(n)$ is a perfect power for all $n\leq 10^8$, and found no other examples.