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Let $a_n$ be the least integer $k$ such that there exist $n$ consecutive integers each with a prime factor $\le k$. For example, $a_{13} \le 11$ because the 13 consecutive integers $114,115,\ldots,126$ each have a prime factor less than or equal to 11. (In fact, $a_{13} = 11$, which is not hard to check.)

Here are the first few values of $a_n$: $$\begin{array}{c | * {20} c} n & 1 & 2 & 3 & 4 & 5 & 6\ldots 9 & 10\ldots 13 & 14\ldots 21 & 22\ldots 25 \\ \hline a_n & 2 & 3 & 3 & 5 & 5 & 7 & 11 & 13 & 17 \end{array}$$

Unless I made a mistake above, this sequence is not in the On-Line Encyclopedia of Integer Sequences. What can be said about it? For example, I am interested in algorithms to compute $a_n$ or estimates of its growth. (Certainly $a_n\le n+1$, as witnessed by the sequence $2,3,\ldots,n+1$.)

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    $\begingroup$ I believe this is closely related to the Jacobsthal function, evaluated at the primorials: oeis.org/A048670 $\endgroup$
    – Gerry Myerson
    Commented Dec 26, 2021 at 23:50
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    $\begingroup$ Ah, thanks. Those are indeed interconvertible, and it looks like the short answers to my questions are "Computation is hard" and "The known upper and lower bounds are far apart". $\endgroup$
    – tuna
    Commented Dec 27, 2021 at 18:56

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