Does the operation $$n \odot m := \prod_{p \text{ prime}} p^{v_p(n) \cdot v_p(m)}$$ on positive integers have a common name? Has this operation been studied somewhere?
Notice that $\odot$ is associative and commutative, but it has no neutral element. It satisfies the distributive law $n \odot (m \cdot k) = (n \odot m) \cdot (n \odot k)$. Two positive integers $n,m$ are coprime if and only if $n \odot m = 1$. If $p$ is a prime number and $k,\ell$ are two integers, then $p^k \odot p^\ell = p^{k \cdot \ell}$.