Consider the two following relations on positive integers $n, m > 0$:
- There exist $i, j > 0$ such that $n^i = m^j$
- There exist $x > 0$ and $i, j > 0$ such that $n = x^i$ and $m = x^j$
It is not difficult to show (using the decomposition of $n$ and $m$ into prime factors) that these two relations are equivalent to one another; and also that they are equivalence relations (they are clearly reflexive and symmetric; the interesting point is that they are transitive).
My question: is there an established name for these relations, or a name for the fact that they are equivalent? My current terminology is to say that "$n$ and $m$ have a common power" (for the first relation), but if there is an established name I'd rather use it.
(My motivation is from word combinatorics, where similar things are true; but since the question already makes sense for natural numbers I thought I'd ask it in this setting.)