Some Special Prime Chains

$\textbf{Question}$: what number begins the longest uninterrupted chain of primes that are either twin, cousin or sexy. In other words, the gap between successive primes must be 2, 4 or 6.

$\textbf{Answer}$: up to ten million the number that begins the longest uninterrupted chain of such primes is, perhaps not surprisingly, $\textbf{3}$. The progression is 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6 and the final prime is 89 after which there is a gap of 8 to the next prime of 97. So between 3 and 89, a gap of 86, there are 23 primes including the first and last primes. It's unlikely that such a concentration of primes will ever occur again amongst the natural numbers but perhaps it's possible. The ratio of 86 to 23 can be expressed as: $$\frac{86}{23}=3.\overline{7391304347826086956521}$$ Thus the average distance between successive primes is slightly under 4 (permalink).

While it's unlikely that this run of 23 twin, cousin or sexy primes will ever be equalled or surpassed what about other runs from 97 upwards of these sorts of primes. A check up to ten million reveals the following record-breaking runs with the starting prime shown:

• 97 with a run of 6 primes
• 149 with a run of 8 primes
• 1277 with a run of 9 primes
• 113143 with a run of 10 primes
• 1464251 with a run of 11 primes

In summary, the sequence is 97, 149, 1277, 113143, 1464251 (permalink).

If we allow runs that equal the previous records then we get the following:

• 97 with a run of 6 primes
• 149 with a run of 8 primes
• 251 with a run of 8 primes
• 587 with a run of 8 primes
• 1277 with a run of 9 primes
• 71327 with a run of 9 primes
• 88789 with a run of 9 primes
• 113143 with a run of 10 primes
• 1464251 with a run of 11 primes
• 7447043 with a run of 11 primes

In summary, the sequence is 97, 149, 251, 587, 1277, 71327, 88789, 113143, 1464251, 7447043 (permalink).