It is well-known how to compute the density of square-free numbers, to get
$$ \lim\_{N\to\infty} \frac{\#\{ n \leq N : n \text{ square-free}\}}{N} = \frac{6}{\pi^2}.$$


What is the density of numbers such that both $n$ and $n+1$ are square-free?
In other words, what is
$$\lim\_{N\to\infty} \frac{\#\{ n \leq N : n(n+1) \text{ square-free}\}}{N} $$
(if the limit exists)?
I'm guessing this has been studied before. Does anyone have a textbook or paper reference?



Density of twin square-free numbers
See *NOTE ON AN ASYMPTOTIC FORMULA CONNECTED WITH r-FREE INTEGERS* by
L. MIRSKY, The Quarterly Journal of Mathematics, Volume os-18, Issue 1, 1947, Pages 178–182, <https://doi.org/10.1093/qmath/os-18.1.178>


This paper is more general, i.e., for $r$ tuples of square free numbers with fixed gap sizes. The number of such integer pairs $\leq x$ is given by 
$$
 Ax+O( x^{\frac{2}{3}+\epsilon}(\log x)^{\frac{4}{3}}),
$$
where $A$ is a constant. See also [here](https://www.hindawi.com/journals/jn/2014/537606/) where the constant $A$ is evaluated in terms of Euler products.


