I'm wondering if there's a term and/or sign already out there for a binary operation on the integers equivalent to $f(x, y) = x / Y$, where $Y$ is the highest power of $y$ that divides $x$, and $y ≠ 0$. I use idempotent in the title to mean that the function $g(x) = f(x, y)$ is idempotent for any nonzero integer $y$. It is equivalent to dividing until there's no division left to be done.
For example (using ⨸ as a placeholder sign): $$ 200 ⨸ 10 = 2,\\ 432 ⨸ 12 = 3,\\ 216 ⨸ 3 = 8,\\ 300 ⨸ 7 = 300 $$
Note that, in the fourth example, $Y = 7^0 = 1$. Or an algebraic example, the equation that got me thinking about this in the first place: $2Z ⨸ 2 = 2Z + 1$, where $Z$ is the set of integers.
If $p$ is prime it's common to write $v_p(n)$ (the $p$-adic order or valuation) for the highest power of $p$ dividing $n$, so that in the special case that you're dividing by a prime we can write
$$f(n, p) = \frac{n}{p^{v_p(n)}}.$$
This operation is pretty poorly behaved if $p$ isn't prime and in that case I'm not aware of any common notation for it. Note that in the prime case $f(n, p)$ is multiplicative in $n$: we have $f(nm, p) = f(n, p) f(m, p)$, which isn't true in the nonprime case.