Find a sequence of 7 consecutive primes

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Find a sequence of 7 consecutive primes, such that these primes have to have the same "gap" in between them. So far I have been doing this in a brute force way, by looking at a list of all the primes and trying different combinations without much luck. Is there a more sophisticated way of doing this?

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http://primerecords.dk/cpap.htm lists largest and smallest known examples of consecutive primes in arithmetic progression of a given length.

The first known sequence is apparently $19252884016114523644357039386451+210n$, but is not known to be the first.