Celebrating Gaps Between Twin Primes

Figure 1

It was back in June of 2017 that I celebrated the end of a drought of twin prime numbers (see post Gaps Between Twin Primes). I'll reproduce here, in Figure 1, the table that appears in that post. On June 22nd, I had just turned 24917 days old and 24917 is a prime that, together with 24919, forms a pair of twin primes. The previous pair is 24419 and 24421. Between the larger of the first pair (24421) and the smaller of the second pair (24917), there is a record gap of 496.

In this gap, there are 40 singleton primes or primes that are not part of a twin prime pair. This is also a record and the first of these singleton primes is 24439 that is a member of OEIS A065044, prime numbers that start a run of exactly \(n\) consecutive primes, none of which are twin primes. This remarkable gap is not surpassed until 62303, a prime that begins a run of 52 singleton primes. As can be seen in Figure 1, these 52 primes lie between the twin prime pairs of (62297, 62299) and (62927, 62929).

Today I turned 26267 days old and this number is also a member of OEIS A065044 marking a run of 36 singleton primes, the last of which is 26669. It needs to be noted that this is not a record run because the sequence registers the first prime in a run of exactly \(n\) primes. Likewise, the gap between the twin primes before and after this run of singleton primes is considerable (418) but far from the record of 496. The twin primes in question are (26681, 26683) and (26261, 26263). Figure 2 shows the situation:

Figure 2: there is a gap of 418 between the twin prime pairs

Here are the members of OEIS A065044 up to \(n\)=40:

2, 47, 113, 79, 2273, 1097, 467, 1327, 1163, 353, 5749, 3011, 5297, 10151, 1493, 9467, 887, 673, 13033, 9049, 15373, 8641, 28759, 83737, 13411, 18553, 14633, 44777, 54037, 60271, 59693, 142169, 77719, 61583, 178939, 26267, 122887, 293269, 89083, 24439

The story is not over yet however, because on the day following this post (when I turned 26268 days old) I discovered that 26268 is a member of OEIS A113274:

A113274

Record gaps between twin primes.

The members of this sequence up to 26268 are as follows:

2, 6, 12, 18, 30, 36, 72, 150, 168, 210, 282, 372, 498, 630, 924, 930, 1008, 1452, 1512, 1530, 1722, 1902, 2190, 2256, 2832, 2868, 3012, 3102, 3180, 3480, 3804, 4770, 5292, 6030, 6282, 6474, 6552, 6648, 7050, 7980, 8040, 8994, 9312, 9318, 10200, 10338, 10668, 10710, 11388, 11982, 12138, 12288, 12630, 13050, 14262, 14436, 14952, 15396, 15720, 16362, 16422, 16590, 16896, 17082, 18384, 19746, 19992, 20532, 21930, 22548, 23358, 23382, 25230, 26268, ...

However, the gap is measured from the smaller of the first twin prime pair to the larger of the second twin prime pair. For example, the gap between (17, 19) and (29, 31) is taken to be 12. Figure 1 shows the gap as being 10 because the gap is measured from the larger of first twin prime pair to the smaller of the second twin prime pair (29 - 19 = 10). Figure 1 is based on OEIS A036063 whose members differ from OEIS A113274 by 2.

A036063

Increasing gaps among twin primes: size.

All the gaps shown in Figure 1 appear in this sequence but the following shows the list of its members extended to 26266:

0, 4, 10, 16, 28, 34, 70, 148, 166, 208, 280, 370, 496, 628, 922, 928, 1006, 1450, 1510, 1528, 1720, 1900, 2188, 2254, 2830, 2866, 3010, 3100, 3178, 3478, 3802, 4768, 5290, 6028, 6280, 6472, 6550, 6646, 7048, 7978, 8038, 8992, 9310, 9316, 10198, 10336, 10666, 10708, 11386, 11980, 12136, 12286, 12628, 13048, 14260, 14434, 14950, 15394, 15718, 16360, 16420, 16588, 16894, 17080, 18382, 19744, 19990, 20530, 21928, 22546, 23356, 23380, 25228, 26266, ...

So the triplet 26266, 26267 and 26268 all make an appearance in this post. The first and last numbers are connected by the fact that the sequences to which they belong are both measuring gaps but in two different ways. The middle number is not connected with its neighbours and is measuring something quite different (the beginning of a record run of singleton primes). It's just coincidence that it falls between its two gap measuring neighbours as it does.