Prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20) are one of the two types of densest permissible constellations of 7 primes. Maximal gaps between septuplets of this type are listed in A201251; see comments and formulas there.
The gap of 83160 between septuplets starting at p=5639 and p=88799 is the very first gap, so a(1)=88799. The gap of 195930 between septuplets starting at p=88799 and p=284729 is a maximal (record) gap - larger than any preceding gap; therefore a(2)=284729. The next gap of 341880 ending at 626609 is again a record, so a(3)=626609. The next gap is smaller, so that gap does not contribute a new term to the sequence.
