exactly divides

Let $a$ and $b$ be integers and $n$ a positive integer.  Then $a^m$ exactly divides $b$ (denoted as $a^n\parallel n$) if $a^n$ divides $b$ but $a^{n+1}$ does not divide $b$.  For example,  $2^4\parallel 48$.

One can, of course, use the similar expression and notation for the elements $a$, $b$ of any commutative ring or monoid (cf. e.g. divisor as factor of principal divisor).