How often is the number between two twin primes divided by 6 a prime?

Question

This question has been edited thanks to the feedback by one user:

12 is in between 11 and 13, and 12/6 = 2 which is prime.

So if we take 29 and 31, 30 is in between, and 30/6=5 which is prime

In general, how often is the number between two twin primes divided by 6 a prime? Are there a finite number or infinite number of them?

Answer 1

With your new edit, asking for prime triplets $(p,6p-1,6p+1)$, this appears in OEIS as sequence A060212.

It is unknown yet whether there are finitely many or infinitely many twin primes. If there were finitely many, then this would obviously imply that there are finitely many prime triplets of the form you are interested in.

It is also unknown whether or not this sequence is infinite.

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A related forum post, giving a 5000+ digit prime in the sequence.

Answer 2

If $p$ and $p+2$ are prime and $p>3$ then $p$, $p+1$ and $p+2 $ are consecutive numbers and $p$ and $p+2$ are not multiple of $3$, so $p+1$ is a multiple of $3$.