"Almost squarefree" numbers?

[Kurt Foster]

If $N$ is a nonzero integer, let core($N$) be the squarefree integer
dividing N such that $N$*core($N$) is a perfect square.  If $|N| > 1$,
the quantity

$f(N) = \ln(N/core(N))/\ln(|N|)$

satisfies $0 \le f(N) \le 1$; $f(N) = 0$ when N  is squarefree, and  
$f(N) = 1$ when $|N|$ is a perfect square.  One can define $N$ as  
being "$\epsilon$-almost squarefree" if $f(N)<\epsilon$.

My question: Is this definition (or something similar) already extant,  
and if so, whose name is attached to it?