From the results posted by Natalia and me, it looks like A113274 is incomplete - missing some terms in the middle.
Second, it is clear that searching for just the largest gap is not going to cut it. If we want to extend the sequence, we want a monotonic sequence of TPT with increasing gap. So that requires one more modification to the App and Asim. Also the largest gap is going to increase forever as the primes get bigger.

Edit: A113274 is not wrong, it deals with not strictly consecutive twin primes. Eg: it can contain (41,43) (59,61) even though there are two primes in between the twins.

I think that the SPT and STPT&TPT applications should work separately, because they will work in different ranges.

Not necessary. I can use batch numbers for the separation. I still need to benchmark it, but the addition of stpt&tpt search has little impact on the speed.

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 59120474339(2): 316 [318]
W 70906853447(2): 340 [342]

W 143334302009(2): 346 [348]

Also found
[209241428699, 209241428701, 209241429071, 209241429073] 372

Thank you for confirming the sequence.
Now you can create a sequence in OEIS.

The "gaps between twin primes" problem is presented in three OEIS sequences

1) https://oeis.org/A113275
A113275 Lesser of twin primes for which the gap before the following twin primes is a record.

2) https://oeis.org/A036063
A036063 Increasing gaps among twin primes: size.

3) https://oeis.org/A113274
A113274 Record gaps between twin primes.

Our sequence can also be represented in three variants.
All data for the three sequences is contained here

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]
. . . . . . .

A sequence similar to A113274 can be called “Rising gaps between of consecutive twin primes”.
This sequence has the form

2, 6, 12, 18, 30, 36, 54, 72, 102, 108, 126, 132, 138, 150, 186, 198, 210, 240, 246, 252, 282, 288, 294, 306, 312, 318, 342, 348, 372

I think it’s not worth creating a sequence similar to A036063, because these are the essence of the reduced by 2 members of the previous sequence

0, 4, 10, 16, 28, 34, 52, 70, 100, 106, 124, 130, 136, 148, 184, 196, 208, 238, 244, 250, 280, 286, 292, 304, 310, 316, 340, 346, 370

A sequence similar to A113275 is, of course, needed.
This sequence has the following form

3, 5, 137, 1931, 2969, 20441, 48677, 173357, 838247, 4297091, 14982551,  30781187, 34570661, 43891037, 79167731, 875971469, 1209266801,
2505898931, 3399081821, 5002002407, 13600912607, 28261373771, 31120423097, 31153871741, 44725020167, 59120474339, 70906853447, 143334302009, 
209241428699

O! I found in OEIS the sequence I made up.
https://oeis.org/A329165
A329165 Let P1, P2, P3, P4 be consecutive primes with P2-P1=P4-P3=2. a(n)=(P3-P1)/6 when the length of the gap with no primes between the two pairs of twin primes sets a record.

1, 2, 3, 5, 6, 9, 12, 17, 18, 21, 22, 23, 25, 31, 33, 35, 40, 41, 42, 47, 48, 49, 51, 52, 53, 57

Why all gaps divided by 6?
Compare with my sequence

2, 6, 12, 18, 30, 36, 54, 72, 102, 108, 126, 132, 138, 150, 186, 198, 210, 240, 246, 252, 282, 288, 294, 306, 312, 318, 342, 348, 372

There is no first term because the first two twins (3,5) and (5,7) intersect.
Next, all the members of my sequence (except 348 and 372) are in the sequence A329165 as a(n)/6.

Until I found the corresponded sequence of twins primes

5, 137, 1931, 2969, 20441, 48677, 173357, 838247, 4297091, 14982551, 30781187, 34570661, 43891037, 79167731, 875971469, 1209266801, 2505898931, 
3399081821, 5002002407, 13600912607, 28261373771, 31120423097, 31153871741, 44725020167, 59120474339, 70906853447, 143334302009, 209241428699

Maybe there is such a sequence too. The search in OEIS is endless!

So, we will continue the sequence A329165. In this sequence, there are now only 26 members.
We have two more members:
a(27) = 58
a(28) = 62.

PS. Entering such a sequence 5, 137, 1931, 2969, 20441, 48677 in the OEIS search field did not find anything.

Batch 55 stpt: testing more of the application. It looks there are still some errors.

I think I solved the last outstanding issue in the application and we can move forwards. This will involve either re-submitting previous regular batch to compare results.
Or move into new area in multiple directions at the same time:
1) continue search for symmetric prime tuples
2) start new search for symmetric and asymmetric twin prime tuples, and twin prime gaps
We could 3) re-submit all of the previous search in the 5e17-6e17 range with the new application: not only that would find the TPT, STPT and twin gaps in the interval, but also verify results from the manual and automatic search at very little additional cost.
There is some work still left on the backend:
4) Benchmark the new app and edit credit multiplier.
5) Fix how twin gap data is inserted into DB. Currently redundant, but not duplicate, entries are inserted.
6) Re-run the assimilator on the whole result database due to format changes.

Batch 56: 530051407000000000 .. 545771407000000000 -1
Count: 8000
Continue with the new application

---

Parameters for further search must be verified or refined. Currently they are:

• smallest even k of SPT to record: 16
  
• smallest odd k of SPT to record: 13 (previously 17)
  
• largest k of SPT: 64 (unchangeable, previously 32)
  
• smallest k of twin prime k-tuple to record: 6
  
• smallest k of symmetric twin prime k-tuples to record: 8
  that is, symmetric consecutive prime tuple composed of 4 twin primes
  
• record gaps between consecutive twin primes
  
• record gaps between consecutive twin primes that are part of STPT k>5
  

No SPT of odd k >= 16 have been found so far. Only in latest batches, I enabled search for k>=13 odd SPTs, and some have been found. We must check OEIS to see if SPTs of smaller odd k are needed.

Besides twin primes, there also exist cousin primes and sexy primes.
Both cousin (d=4) and twin (d=2) do not intersect or touch themselves greater than 7. My program exploits this property for twin primes. It could be easily extended to look for cousin prime tuples as well. But sexy primes (d=6) are harder, since they can can form up to quadruplets. Also when we start looking for cousin prime tuples, or sexy prime tuples, we could start looking for various combinations of twin and cousin primes and that will get messy quickly. Is there a motivation?

Batch 57: 545771407000000000 .. 577211407000000000 -1
Count: 16000
Continue with the new application

---

I think we need to limit ourselves to finding twins.

Where can I see the results for the twins for k=6 (12-tuple) and k=7 (14-tuple)?
I found only this
https://boinc.tbrada.eu/spt_list.php?k=12

. . . . . . .
999475912613: 0 26 38 66 68 96 110 138 140 168 180 206
999476523211: 0 12 18 22 36 52 96 112 126 130 136 148
999723054251: 0 6 26 36 48 60 98 110 122 132 152 158
999801043721: 0 2 12 32 72 78 110 116 156 176 186 188
999930089267: 0 2 20 32 44 54 62 72 84 96 114 116
999933514397: 0 12 30 60 110 114 152 156 206 236 254 266
999938914729: 0 10 24 60 82 84 100 102 124 160 174 184
999954513763: 0 18 58 60 76 148 168 240 256 258 298 316
999982936349: 0 14 18 50 68 78 80 90 108 140 144 158
532905152025296537: 0 2 54 56 60 62 84 86 90 92 144 146
533308875499369997: 0 2 24 26 30 32 54 56 60 62 84 86
541024702436824187: 0 2 30 32 114 116 150 152 234 236 264 266
541267134297120527: 0 2 30 32 72 74 132 134 174 176 204 206
# last = 331678 # count = 15036

I quickly created two pages, but I am working on better page to explore the database.
https://boinc.tbrada.eu/spt_list_stpt.php?k=8 - symmetric tuples of twin primes (k is number of primes)
https://boinc.tbrada.eu/tpt_list.php?k=9 - asymmetric tuples of twin primes (k is number of pairs)
All STPTs are included in SPT list as well.

# Copyright Tomas Brada, ask on forum about reuse or citation.
# where kind = tpt
# where k = 9
532359367138961447: 0 2 42 44 72 74 114 116 192 194 252 254 330 332 402 404 444
534790455001337861: 0 2 216 218 240 242 258 260 300 302 306 308 366 368 396 398 516
535474368412823759: 0 2 162 164 348 350 372 374 390 392 402 404 540 542 570 572 672
536706305502981749: 0 2 78 80 120 122 282 284 438 440 510 512 522 524 588 590 672
539203083290572667: 0 2 30 32 42 44 72 74 150 152 324 326 390 392 414 416 432
541603519435938941: 0 2 48 50 78 80 126 128 288 290 300 302 330 332 456 458 480
545690807892680447: 0 2 30 32 84 86 90 92 180 182 312 314 324 326 450 452 504
574215423057543257: 0 2 12 14 102 104 144 146 180 182 222 224 282 284 312 314 324
574866882160551047: 0 2 12 14 42 44 60 62 84 86 90 92 114 116 240 242 270
576401886649823237: 0 2 42 44 90 92 120 122 150 152 222 224 390 392 432 434 474
576790168452150197: 0 2 42 44 54 56 84 86 240 242 270 272 450 452 462 464 492
# last = 486160 # count = 11

Excellent!
Please see these sequences in OEIS

https://oeis.org/A087641

A087641 Start of the first sequence of exactly n consecutive pairs of twin primes.
29, 101, 5, 9419, 909287, 325267931, 678771479, 1107819732821, 170669145704411, 3324648277099157, 789795449254776509

https://oeis.org/A259034

A259034 Start of a string of exactly 9 consecutive (but disjoint) pairs of twin primes.
170669145704411, 597655503030737, 1209758169609917, 1529543606818727, 1980326398382819

We have for k=9 a large missed interval (1980326398382819, 532359367138961447).

Similarly for k=8.
See
https://oeis.org/A263205

A263205 Start of a string of exactly 8 consecutive (but disjoint) pairs of twin primes.
1107819732821, 3735283249697, 4588646146631, 6340698579419, 8412649748537, 9206359843907, 9667145661911, 10261787848841, 10877306469737, 13792968231041, 17231043159311, 18996369140627, 21471510972419, 21791129807147, 23105869316669, 23224938371519

The reason there are no new tasks for SPT, is because there is a problem with STPT detection code.
I identified the problem, but need to verify that it works and does not affect anything else.

Batch 58: 577211407000000000 .. 600001477000000000 -1
Count: 11598
Continue with the new application

---

There is a very large number of STPT(8) (4 pairs), over 2 million and it was slowing the assimilator quite a lot. So, I disabled saving those into the tuple database. The result files, remain intact and archived for later use.

List of OEIS sequences for new search interval:
https://oeis.org/A087641 TPT(N)
https://oeis.org/A035795 TPT(7)
https://oeis.org/A263205 TPT(8)
https://oeis.org/A259034 TPT(9)
https://oeis.org/A274792 STPT(n)
https://oeis.org/A330278 STPT(12)
https://oeis.org/A175309 SPT(N) ?
https://oeis.org/A055382 ...?
https://oeis.org/A113274, https://oeis.org/A113275 twin_gaps
https://oeis.org/A329164, https://oeis.org/A329165 twin_gaps
https://oeis.org/A256234 4x4


Missing sequences for STPT and SPT. I will add later.

Batch 59: 5 .. 1000000000005 -1
Count: 100
Scan beginning with new App.

inp.mine_k= 14;
inp.mino_k= 9;
inp.max_k= 64;
inp.upload = 0;
inp.exit_early= 0;
inp.out_last_primes= 0;
inp.out_all_primes= 0;
inp.primes_in.clear();
inp.twin_k=6;
inp.twin_min_k=8;
inp.twin_gap_k=255;
inp.twin_gap_min=249;
inp.twin_gap_kmin=75;
wu.delay_bound = 4 * 3600;

---

We have the sequence https://oeis.org/A113274 and the corresponding sequence https://oeis.org/A113275.
We have the sequence https://oeis.org/A329165, but we do not have the corresponding sequence with real values of the twins primes.

I found the first terms of this sequence

5, 137, 1931, 2969, 20441, 48677, 173357, 838247, 4297091, 14982551, 30781187, 34570661, 43891037, 79167731, 875971469, 1209266801, 2505898931, 3399081821, 5002002407, 13600912607, 28261373771, 31120423097, 31153871741, 44725020167, 59120474339, 70906853447, 143334302009, 209241428699

Tomáš Brada
please create this sequence first.
Indicate authorship and the link to the results
https://boinc.tbrada.eu/forum_thread.php?id=3055&postid=3915#3915

Then there will be the results of the TBEG project.

PS. I did not find such a sequence in OEIS. But maybe I was looking badly?