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?


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 where the constant $A$ is evaluated in terms of Euler products.

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