I wonder if anyone has done any systemic work on gap symmetry? By this I mean there are consecutive gaps that are symmetrical in length around a central point. This is equivalent to saying there are primes symmetrically distant from a central point.

Pioneer work has been done - and the central point has been dubbed "interprime"

The simplest is defined half way between any two primes - each prime is equidistant from the central point. This series of numbers is defines as (p(1)+p(2))/2, and this is listed on OEIS as interprimes -

https://oeis.org/A024675
For two primes either side of the central point the series is shown as

https://oeis.org/A263674 double interprimes 9,12,15,18,30,42... where, for example primes 5 and 7 are 4 and 2 distant from 9, as are 13 and 11

For three primes either side, the series is 12,15,30,42,105,144... where, for example primes 41,37 and 31 and 1,4, and 11 away from 42, as are 53,47 and 43 . This series is not listed on Sloane

The list for 4-interprimes looks quite odd! 15645,19425,34485,34845,35988,46641...

5-interprimes..783630 is the only one up I could find up to pi(100000). Consecutive nearby primes are 783569 (61 away), 783571 (59 away), 783599 (31 away), 783613 (17 away), 783619 (11 away), 783641 (11 away), 783647 (17 away), 783661 (31 away), 783677 (59 away) and 783689 (61 away). The gap sequence is 2,28,14,6,22,6,14,28,2

Dirichlet I'm sure gives us the insight that there are infinitely many n-interprimes for any value of n.

Current computing power would suggest that it should be possible to find 10-interprimes (i.e. a list of 20 primes)

I also noted that although there are 70 or so lists of interprimes on OEIS, none look at primorials and multiples of primorials. I would have thought these would have provided a good grounding for finding n-interprimes, given their modular symmetry around the centre point. "close to" primorials i.e. p#/x, where x is a squarefree p-smooth number are also good hunting ground.

As examples:

the 3-interprime series listed above 12,15,30,42,105,144,165,312 could be restated as:

2*3#... 5#/2...5#...7#/5...7#/2...24*3#...11#/14... 52*3#.

However, near primorials are less impressive for the 4-interprime list:

15645,19425,34485,34845,35988,46641 could be restated as:

149*7#/2...185*7#/2...209*11#/14..2323*5#/2...5998*3#...2221*7#/10