I found many primes sequence with pattern like $Q=\{p ~{\rm ∈~ prime}~|~p,p+30,p+60,p+90,p+120,p+150\}$, is there exist infinite prime $p$ holds such sequence? Then the elements in set $~Q~$ are primes ( in arithmetic progression), such as when $p=541,2221,135151,279421, 10^{20}+3346371,10^{40}+503428698001$ (the two large primes are Peter's finding), .
Below tables include some findings:
This is another pattern $~{Q=\{p∈prime |p,p+12,p+24,p+36}\}$
