Let $q_1,q_2,\ldots$ denote the squarefree integers 1, 2, 3, 5, .... What effective bounds are known for $q_n$? Clearly $$q_n\sim\zeta(2)n$$ but I need hard inequalities. Of course from the above there exist $\varepsilon,N$ with $$(\zeta(2)-\varepsilon)n < q_n < (\zeta(2)+\varepsilon)n$$ for all $n>N,$ but I do not have proven values for $\varepsilon,N$. Of course $$|q_n-\zeta(2)n| < f(n)$$ for sublinear $f$ would be preferable (and should be possible; squarefree numbers are fairly well-behaved).
I'm sure this is in some standard reference but I haven't found it. Ideas?
For those interested, my actual goal is to find a reasonable bound for the powerful (2-full) numbers for computational purposes. Their asymptotic growth is tightly constrained but for numerical computations I would prefer worst-case bounds that I can trust rather than a heuristic that says the error probably won't be much more than, say, 10 times the O-term.