The Online Encyclopedia of Integer Sequences has a nice page on the gaps found related to the Jacobsthal function for primorials.
I would like to take a look at the sequences that have these gaps shown in this list. Using a brute force algorithm, I was able to find the the start of $n=9$ the gap for $23\#$ which starts at $20332472$.
I have an algorithm which can quickly find gaps of length $2(p_{n-1}+1)+2$ which works for $n=1$ thru $n=8$ and also includes $n=10, 11, 13$.
Is there a list in oeis that lists the start of these gaps? I was not able to find it. If not, does anyone know of such a list? I would be very interested in seeing the patterns that can be found in these sequences of consecutive integers.
A list, up to $24$, is given at https://oeis.org/A049300/b049300.txt
Here it is: $$ \matrix {1& 2\hfill\cr 2 &2\hfill\cr 3& 2\hfill\cr 4& 2\hfill\cr 5 &114\hfill\cr 6 &9440\hfill\cr 7 &217128\hfill\cr 8 &60044\hfill\cr 9& 20332472\hfill\cr 10& 417086648\hfill\cr 11& 74959204292\hfill\cr 12 &187219155594\hfill\cr 13 &79622514581574\hfill\cr 14 &14478292443584\hfill\cr 15 &6002108856728918\hfill\cr 16 &12288083384384462\hfill\cr 17 &5814429911995661690\hfill\cr 18 &14719192159220252523420\hfill\cr 19 &7714600835154917969172\hfill\cr 20 &396499427735806558888610\hfill\cr 21 &3084626641924131277081102880\hfill\cr 22 &77624949518856340312883476188\hfill\cr 23 &23314635458774315797705662793622\hfill\cr 24 &990578949024486399796411481983580\hfill\cr} $$