Quote:
Originally Posted by Dr Sardonicus
One observation: If N is an interprime, N is not divisible by 3, and the smallest prime in the interval is greater than 3, then all the gaps must have length divisible by 3.
For if N =/= 0 (mod 3), p is one of the primes, p =/= N (mod 3), and p =/= 0 (mod 3), then the symmetrically located number 2*N  p will be == 0 (mod 3).
This condition might narrow things down a bit.

Neat! For 3565765770 with prime factors 2,5,356576577 the gaps are 6,6,24,6,18,18,18,6,24,6,6, all 0mod3.
It is easy to demonstrate in general that the majority of gaps are not 0mod3 in length  for example in the first 100000 gaps, the sum of 1mod3 and 2mod3 gaps is about 38.4% more than 0mod3  and hence the longer (larger n) the ninterprime, the greater the requirement for all prime gaps involved in the interprime to be 0mod3, and subsequently the less likely the interprime is 1 or 2mod3. Using the 41.9% 0mod3 gap density in the first 100000 gaps as a given, the chances of 11 gaps in a row 0mod3 is about 14157:1  this would provide for less 1mod3 and 2mod3 6interprimes than are actually turning up 6 out of 896 is 149:1  this could of course just be chance  but I wonder if I am again missing something here.
Does the ratio of the count of successive prime gaps (1mod3+2mod3)/0mod3 tend towards a constant? Can't see why not. Where can I look for work done on that?
I've tested up to 1.5e10 and, in addition to the 9interprime mentioned above, there are two others:
74422046685
81661695390
There are 33 8interprimes in the range 0 to 1.5e10
But no 10interprime yet