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Erdős problem 451

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Let nkn_k be the least integer greater than 2k2k for which ∏i=1k(nk−i)\prod_{i=1}^k (n_k - i) has no prime factor in (k,2k)(k, 2k). How rapidly must nkn_k grow?

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Let $n_k$ be the least integer greater than $2k$ for which $\prod_{i=1}^k (n_k - i)$ has no prime factor in $(k, 2k)$. How rapidly must $n_k$ grow?

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