Greetings Koichi Ohno,
I have been learning about prime numbers for the last two years. I have focused on the (6n-1) (6n+1) pair relationship, and I have found some similar information.
Have you considered the ratios of Primes to all numbers in this subset
The same ratio for Twin Primes and Double Twin Primes are interesting.
If you are interested, I have used this and other patterns that I have to develop an efficient sieve.
One of the places I have discussed this idea is here:
https://coderanch.com/t/791015/languages/Prime-Number-Sieve-Idea
I would be interested to hear what you think about these other patterns.
Bill Donelson