todo: check this: W 525725685315463739(6): 10 16 88 16 10

See https://boinc.progger.info/odlk/forum_thread.php?id=49&postid=5036#5036

525725685315463669: 0 10 70 72 82 84 100 102 190 192 208 210 220 222 282 292
527429414970847919: 0 14 48 50 78 80 102 104 138 140 162 164 192 194 228 242
528735672273276389: 0 54 60 62 78 80 102 104 138 140 162 164 180 182 188 242
529528572209967023: 0 18 66 68 78 80 108 110 126 128 156 158 168 170 218 236

Also

U 525725685315463669(16): 10 60 2 10 2 16 2 88

OK. I verified that the tuple is identified correctly. We are dealing with group of 6 twin primes, which are also symmetric (12-tuple) and a part of larger symmetric 16 tuple.

U 525725685315463669(16): 10 60 2 10 2 16 2 88

Please save the usual format in the database for 16-tuples

525725685315463669: 0 10 70 72 82 84 100 102 190 192 208 210 220 222 282 292

There are symmetric tuples which are only partly made from twins. The question is how to present them? For example:

525725685315463669: 0 10 70 72 82 84 100 102 190 192 208 210 220 222 282 292

It's clear that this is 16-tuple, so it is listed with other 16-tuples. It will be also reported as symmetric group of 6 twins starting at 525725685315463739.
The question is, should there be a note that the group of 6 twins is part of larger tuple?

The question is, should there be a note that the group of 6 twins is part of larger tuple?

No.

Batch 54: symmetric twin prime tuples - testing the application.
The server side software is missing for the new fields, so the validator is disabled.
Thanks for the reply Natalia.

First result from the new application:

largest_twin_gap: 525725689650797957 490

The gap between those two primes is larger than one in your message.
Expanded: 525725689650797957 525725689650797959 525725689650798449 525725689650798451.

Expanded: 525725689650797957 525725689650797959 525725689650798449 525725689650798451.

Yes, this is a new record.

To all

https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;6a849ab3.9811
https://oeis.org/A087641

New record: 500306885940569681 724
I am just looking thru the result files on the server, because validator is not yet working. I see 16 out of 105 corrupted files, which suggests that there is a error in the app.

New record: 500306885940569681 724

Dear participants!
Maybe you know other records or articles that describe this problem.
Please inform.
I did not find on the Internet.
Perhaps the Prime Grid project solves this problem.

There is a page on PolyMath related to prime gaps. But it is an other problem. The keyword "prime gaps" can lead to more results.

Please look this
Maximal gaps between twin primes up to 1.2 × 10^15
https://javascripter.net/math/primes/maximalgapsbetweenktuples.htm#2tuples

But are there twins from consecutive primes (other than those at the beginning)?

1st twin pair: 2nd twin pair: Gap g2(p):
3 5 2
5 11 6
17 29 12
41 59 18
71 101 30
311 347 36
347 419 72
659 809 150
. . . . . . . . .

74668675816277 74668675834661 18384
116741875898981 116741875918727 19746
136391104728629 136391104748621 19992
221346439666109 221346439686641 20532
353971046703347 353971046725277 21930
450811253543219 450811253565767 22548
742914612256169 742914612279527 23358
1121784847637957 1121784847661339 23382
1149418981410179 1149418981435409 25230

See also in OEIS
https://oeis.org/A113274

PS. Who can quickly check if there are twins from consecutive primes in the list shown?

Note: the gap between the two twins primes here is calculated as the difference between the first numbers of twins.

In our example

1st twin pair: 500306885940569681, 500306885940569683
2nd twin pair: 500306885940570407, 500306885940570409
Gap g2 (p) = 500306885940570407 - 500306885940569681 = 726

This does not change the essence of the largest gap between twins of consecutive primes.

I started from scratch and got the following gaps for the twins of their consecutive primes

[3, 5, 5, 7] 2
[5, 7, 11, 13] 6
[137, 139, 149, 151] 12
[1931, 1933, 1949, 1951] 18
[2969, 2971, 2999, 3001] 30
[20441, 20443, 20477, 20479] 36
[48677, 48679, 48731, 48733] 54
[173357, 173359, 173429, 173431] 72
[838247, 838249, 838349, 838351] 102
[4297091, 4297093, 4297199, 4297201] 108
[14982551, 14982553, 14982677, 14982679] 126
[30781187, 30781189, 30781319, 30781321] 132
[34570661, 34570663, 34570799, 34570801] 138
[43891037, 43891039, 43891187, 43891189] 150
[79167731, 79167733, 79167917, 79167919] 186
[875971469, 875971471, 875971667, 875971669] 198
[1209266801, 1209266803, 1209267011, 1209267013] 210
[2505898931, 2505898933, 2505899171, 2505899173] 240
[3399081821, 3399081823, 3399082067, 3399082069] 246
[5002002407, 5002002409, 5002002659, 5002002661] 252
[13600912607, 13600912609, 13600912889, 13600912891] 282

Am I not mistaken?

Is there such a sequence in OEIS?
I'll take a look now.

I entered the search sequence 2, 6, 12, 18, 30, 36, 54, 72, 102, 108
That's what is reported

Search: seq:2,6,12,18,30,36,54,72,102,108
Sorry, but the terms do not match anything in the table.

If your sequence is of general interest, please submit it using the form provided and it will (probably) be added to the OEIS! Include a brief description and if possible enough terms to fill 3 lines on the screen. We need at least 4 terms.

There seems to be no such sequence.
If I correctly calculated the gaps, a sequence can be created (similar to the sequence https://oeis.org/A113274).
Then the sequence can be continued.

Unfortunately, I cannot create sequences in OEIS, as I is blocked.

Tomáš Brada
I suggest you check my calculations and create a sequence in OEIS.
Then you will continue this sequence in your project.

I also found solutions

[28261373771, 28261373773, 28261374059, 28261374061] 288
[31120423097, 31120423099, 31120423391, 31120423393] 294
[31153871741, 31153871743, 31153872047, 31153872049] 306

I am very interested in these results.

The program on PARI/GP is slow.

I will send a workunit(s) to enumerate all k>=2 k-tuples of twin primes up to some comfortable number.

The asimilator has been adapted for the new app and batch 53 has been validated. Yes, there were some glitches, if you are going to ask about the strange wu status. Both SPT, STPT and TPT are now in database. The spt_list.php page shows only spt/stpt for now. The current largest gaps are:

+--------------------+-----+---+-----------------+
| start              | d   | k | ofs             |
+--------------------+-----+---+-----------------+
| 500307392152057409 | 232 | 5 |  76 28 22 4 232 |
| 500306885940569681 | 724 | 0 |                 |
+--------------------+-----+---+-----------------+

(first row is from 1.st approach, second is 2.nd approach)

I think that the SPT and STPT&TPT applications should work separately, because they will work in different ranges.
The SPT application continues the range from 5.3*10^17.
STPT&TPT applications must start earlier.

Here is sequence of ever increasing gaps between consecutive twin pairs.
Number 3 is excluded, because my program works only for p>=5. The gap, as a difference between first prime of the twin, is in square brackets.

twin_gap_seq:
W 5(3): 4 4 [6]
W 137(2): 10 [12]
W 1931(2): 16 [18]
W 2969(2): 28 [30]
W 20441(2): 34 [36]
W 48677(2): 52 [54]
W 173357(2): 70 [72]
W 838247(2): 100 [102]
W 4297061(3): 28 106 [108]
W 14982551(2): 124 [126]
W 30781187(2): 130 [132]
W 34570661(3): 136 46 [138]
W 43891037(2): 148 [150]
W 79167731(2): 184 [186]
W 875971469(2): 196 [198]
W 1209266801(2): 208 [210]
W 2505898931(2): 238 [240]
W 3399081821(2): 244 [246]
W 5002002407(2): 250 [252]
W 13600912607(2): 280 [282]
W 28261373771(2): 286 [288]
W 31120423097(3): 292 64 [294]
W 31153871741(2): 304 [306]
W 44725020167(2): 310 [312]
W 59120474339(2): 316 [318]
W 70906853447(2): 340 [342]

W 143334302009(2): 346 [348]