> there's also the dubious, or trivial, or dunno (gotta generalize this pattern as well) of the first "consecutive" twin prime but they overlap which is 3,5 and 5,7.... which reminds me of how only 2 and 3 are both primes off by one; again, I need to generalize this pattern of "last time ever primes did that"

For the triplet n, n+2, n+4, exactly one of those numbers is divisible by 3. So the only triplet n, n+2, n+4 where all numbers are prime contains 3: 3, 5, 7.