Powerful-squarefree coprime decomposition

Every positive integer has a unique factorization $n=as$ with $a$ powerful, $s$ squarefree, and $\gcd (a,s)=1$: assign each prime power of exponent at least two to $a$, and each prime of exponent one to $s$. In general $s$ differs from the squarefree part, which instead retains every prime of odd exponent.