Problem #386: Let $2\leq k\leq n-2$

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Progress

Erdős and Graham write that 'a proof that this cannot happen infinitely often for $\binom{n}{2}$ seems hopeless; probably this can never happen for $\binom{n}{k}$ if $3\leq k\leq n-3$.'

Weisenberg has provided four easy examples that show Erdős and Graham were too optimistic here:\[\binom{7}{3}=5\cdot 7,\]\[\binom{10}{4}= 2\cdot 3\cdot 5\cdot 7,\]\[\binom{14}{4} = 7\cdot 11\cdot 13,\]and\[\binom{15}{6}=5\cdot 7\cdot 11\cdot 13.\]The known values of $n$ for which $\binom{n}{2}$ is the product of consecutive primes are $4,6,15,21,715$ (see A280992).