**Question:** Is there a function $f(n) \rightarrow \infty$, such that infinitely often the interval $[n,n+\frac{f(n) \log(n)}{\log{\log(n)}}]$ does not contain a squarefree integer?

**Additional information:** If we consider the simultaneous congruences:

$$ x \equiv 0 \ (\text{mod} \ 2^2)\\ x \equiv 1 \ (\text{mod} \ 3^2) \\ \cdots \\ x \equiv k-1 \ (\text{mod} \ p_{k}^2) $$

Which has a solution not bigger than $ \Pi_{i=1}^{k} p_i^2 \leq e^{2(1+\varepsilon)k\log k} $.

This way we can see that there is a constant such that there are no squarefree numbers in the interval $[n,n+c\frac{\log(n)}{\log{\log(n)}}]$ infinitely often. We can optimize the above equations, for example on the right hand side we don't have to consider numbers divisible by $4$ after the first equation. Or numbers congruent to $1$ modulo 9 after the second equation etc. but that just gives a better constant!

muchbigger than your lower bound. Granville showed that the abc conjecture implies an upper bound of $n^\epsilon$ for any $\epsilon>0$. $\endgroup$ – Anthony Quas Nov 18 '13 at 2:13