All Random Solved Random Open
Let $r\geq 0$. Does the density of integers $n$ for which $\binom{n}{k}$ is squarefree for at least $r$ values of $1\leq k<n$ exist? Is this density $>0$?
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.