Prime-support equivalence relation ($m\sim n⟺\mathrm{rad}(m)=\mathrm{rad}(n)$)

Positive integers are equivalent when they have the same finite set of prime factors. Equivalently, each divides a positive power of the other. This equivalence relation is induced by the radical of an integer; its equivalence classes correspond to finite prime supports. The empty support gives the singleton $\{1\}$, and every nonempty support gives an infinite class by varying exponents.