I'm looking for a way to calculate [OEIS A144311](https://oeis.org/A144311) efficiently.

In one sense or another, this series considers the number between "relative" twin primes. What do I mean by this?

Well, the number $77$ is relatively prime to {$2,3,5$} even tho it's not actually prime. We're looking for relative twin primes to this set tho: consecutive odds relatively prime to {$2,3,5$}. For instance, $(77,79)$ would be an example of such. Neither number is divisible by {$2,3,5$}.

$78$ then is the number between these. The next smallest number between a similarly relative twin prime pair would be $90$. $(89,91)$ is the next relative (and real) twin prime pair. That gives us a relative twin prime gap of 12 (being $90-78$), and 11 fails between successes. The question is, what is the largest possible twin prime gap relative to just {$2,3,5$}? This so happens to be it. 11 is the max fails for {$2,3,5$}. The series says that 29 is the max fails for {$2,3,5,7$}, and 41 for {$2,3,5,7,11$}, the first 5 primes.

Now that we've well defined these, what would be the best way to calculate them? Also, what is the 17th number in this series? All methods I've used to try to calculate these have resulted in numbers beyond what can be handled well even in something like Mathematica even by the 10th number in the series.