Leading to the question:
A prime range must be even.
In a a range between $2$ numbers, there can be only $2$ possible consecutive primes.
In the range between $4$ numbers, there are only $3,5,7$ that are $3$ consecutive primes, but other than that there could be only a max combination of $2$ consecutive primes.
In the range between $6$ numbers, there can be only a max combination of $3$ consecutive primes.
In the range between $8$ numbers, there can be only a max combination of $3$ consecutive possible primes.
...
Data out of the first million primes shows that:
So far all these possible combinations of consecutive primes within a range seem to be distributed evenly throughout infinity (twin primes, cousin primes, sexy primes, twin primes followed by cousin primes, cousin primes followed by twin primes...)
When mentioning “distributed evenly”, I mean that there is more or less a way (range) to predict the “next” occurrence.
I was trying to find ranges in which the combinations are not distributed evenly and the first small case I found is the following:
In the case of the range of $16$, whether there could be a combination of 6 primes is definitely permitted, but out of the first million primes doesn't seem to be distributed evenly:
Out of the first million primes there were only $6$ cases. the first $5$ $ \leqslant 43793$ and the last one much much further at $1091273$ not seen again (the millionth prime is $1299709$. and from there on I have no idea ...)
Here are all the $6$ cases: $7-23 , 97-113 , 16057 - 16073 , 16057 - 16073 , 19417 - 19433 , 43793 - 43777 , 1091257 - 1091273$
Am I not seeing the even distribution because 1 million primes is too small of a test range?
In theory do all permitted range/primes combination go on to infinity? or is it possible to have a finite limit?