There are many results about consecutive integers all having small prime factors. But what about consecutive integers each of which has a large prime factor?

More precisely, let $P(n)$ be the greatest prime factor of $n$.

Is it true that for each $k \geq 1$ there exists $n \geq 1$ such that $P(n + i) > \sqrt{n + i}$ for all $i=1,\dots,k$ ?

If yes, what is an upper bound in terms of $k$ for the least possible $n$ ?

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