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Start with prime $p$ then find the maximum sequence of primes $p+gap, p+2\cdot gap, p+3\cdot gap$ etc. Looking at the first gaps I found: \begin{array}{|c|c|} \hline 2 & 1 \\ \hline 4 & 2 \\ \hline 6 & 5 \\ \hline 8 & 2 \\ \hline 10 & 2 \\ \hline 12 & 5 \\ \hline 30 & 6 \\ \hline 210 & 10 \\ \hline 2310 & 9 \\ \hline 30030 & 12 \\ \hline \end{array}

I checked all gap sequences with $p$ up to the first $100000$ primes.

I noticed two things:

1) Most Multiples of these gaps seem to have the same maximum. For example 4 and 8, 6 and 12, but also 30 and 60.

2) There is a connection to the primorials so I also added the primorial gaps. That was a success up to 210 but then the maximum dropped for 2310 and then increased again to reach a maximum of 12 for 30030.

Questions:

1) Is there a gap that produces more than 12 results?

2) How far should I check the prime $p$ for a gap to ensure that there is no new maximum sequence?

3) Is there information on research already done or existing theories for this problem?

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