According to a conjecture there are no three consecutive powerful numbers.
Necessary condition for this is integer solution of
$$ z^3 y^2 = x(x-1)(x+1) \qquad (1) $$
What are integer solutions of (1)?
For fixed $z$ Weierstrass model is
$$ v^2 = u^3 - z^6 u$$
$x = u/z^3, y= v/z^6$. Since $z$ is integer $u,v$ must be integers too.