I understand that the maximal distance between consecutive coprimes to a primorial $P_n\#$ has order $O(P_n^2)$. My question is, given that you can ONLY use prime numbers less than X and no others, is the maximum number of consecutive integers with those prime factors the smallest prime $p>X$ minus 2? It seems like this would only occur at the beginning of the number line, but I also have my doubts on that, just mentioning for conversation. The coprime problem assumes any primes can be used as long as each consecutive number contains at least one prime less than or equal to $P_n$. Any thoughts are welcome. References to papers or proofs would be awesome. Thanks!
I think I figured it out. Say $p$ is the smallest prime larger than $X$. Since multiples of $p$ occur every $p$ units, you cannot have a string of consecutive integers longer than $p$ that does not contain a multiple of $p$. Hence, $p$ is an upper bound on the maximum number of integers containing ONLY prime factors less than $X$.