Problem #378: Let $r\geq 0$. Does the density of integers $n$ for which...

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Progress

Erdős and Graham state they can prove that, for $k$ fixed and large, the density of $n$ such that $\binom{n}{k}$ is squarefree is $o_k(1)$. They can also prove that there are infinitely many $n$ such that $\binom{n}{k}$ is not squarefree for $1\leq k<n$, and expect that the density of such $n$ is positive.

Aggarwal and Cambie have observed this problem is resolved by the results of Granville and Ramaré [GrRa96], who in particular show that the density of the set of those $n$ such that $\binom{n}{k}$ is squarefree for exactly $2m+2$ many values of $k$ exists. If this density is $\eta_m$, then the density in the original question is simply\[1-\sum_{0\leq m\leq \frac{r-1}{2}}\eta_m.\]This density is positive since $\eta_{r+1}>0$.