Is there a name for this type of integer?

Question

An integer $n$ such that $\exists$ at least one prime $p$ such that, $p|n$ but $p^2$ does not divide $n$.

i.e. : an integer with at least one prime that has a single power in the prime factorization.

Do these numbers have a special name, and have they been studied?

Answer 1

The numbers not in your set are called the powerful numbers (There are other names.)

I do not know of a special name for the complement of the set of powerful numbers. The powerful numbers have been studied. Please see the above link for a start.

Answer 2

These are sometimes called weak numbers, presumably because not weak naturals are precisely the powerful naturals. However the weak terminology does not appear to be anywhere near as widely used as is the powerful terminology (e.g. Granville and Ribenboim use "not powerful"). I also recall seeing other names besides weak (e.g. impotent).