Triplets of Disjoint Twin Primes

Today, having turned 26861 days, I noticed that it formed the smaller of a pair of twin primes but more than that it had a special property that qualified it for membership of OEIS A035791:

A035791

Start of a string of exactly 3 consecutive (but disjoint) pairs of twin primes.

The requirement that the three pairs be disjoint disqualifies prime pairs like (101, 103), (103, 107) and (107, 109) because of the 103 and 107 overlap. It turns out that such triplets of twin primes are relatively rare. The previous was 21587 and the next will be 49367! 

I managed to get SageMathCell to generate the OEIS sequence up to 100,000,000 as well as a table showing the record gaps between the first prime of the first prime pair and the last prime of the last prime pair. The table is shown in Figure 1 below in the range up to 100,000,000:

Figure 1: permalink

As can be seen, 26861 holds the equal record gap of 32 with the next member (49367) creating a new record of 44. The members of the sequence up to one million are shown below:

5, 179, 809, 3359, 4217, 6761, 9419, 9431, 18041, 21587, 26861, 49367, 62969, 62981, 67187, 72221, 72227, 80447, 82721, 91127, 97841, 98897, 103967, 109829, 122597, 154157, 178037, 203321, 208931, 225749, 227609, 236867, 243671, 251201, 266447, 285611, 289109, 295871, 317729, 330287, 342047, 358877, 375251, 392261, 392267, 397517, 402329, 405047, 420809, 422087, 440549, 444341, 452519, 489911, 495569, 495587, 524507, 524969, 560477, 563411, 565889, 570497, 595139, 622187, 629567, 632297, 636059, 640229, 641519, 651179, 663569, 663581, 670037, 677459, 686009, 690839, 704549, 746507, 753437, 758711, 768167, 773609, 773951, 777389, 795761, 797567, 830309, 831371, 842321, 854897, 873569, 875261, 875981, 907367, 909287, 909299, 909317, 946079, 983429, 994307, 997811

The full range of OEIS sequences are:

• A035789: Start of a string of exactly 1 consecutive (but disjoint) pairs of twin primes.
• A035790: Start of a string of exactly 2 consecutive (but disjoint) pairs of twin primes.
• A035791: Start of a string of exactly 3 consecutive (but disjoint) pairs of twin primes.
• A035792: Start of a string of exactly 4 consecutive (but disjoint) pairs of twin primes.
• A035793: Start of a string of exactly 5 consecutive (but disjoint) pairs of twin primes.
• A035794: Start of a string of exactly 6 consecutive (but disjoint) pairs of twin primes.
• A035795: Start of a string of exactly 7 consecutive (but disjoint) pairs of twin primes.

For quadruplets and beyond, the numbers are very large except for the first few members of OEIS A035792 (quadruplets) that are below a million: 9419, 62969, 72221, 392261, 495569 and 663569.