Maximal gaps between prime triplets

Prime triplets (3-tuples) are the densest permissible clusters of 3 consecutive primes. There are two types of prime triplets:

{p, p+2, p+6} (OEIS A022004, A201598, A201599, A233434)

{p, p+4, p+6} (OEIS A022005, A201596, A201597, A233435).

The observed maximal gaps between prime triplets near p are at most log p times the average gap.
The approximate size of a maximal gap that ends at p is given by the following empirical formula:

E(max g₃(p)) = a(log(p/a) − 2/3) = O(log⁴p)

where a = 0.34986 log³p is the average gap, as predicted by the Hardy-Littlewood k-tuple conjecture.

Maximal gaps between prime triplets of each type are listed below.

Maximal gaps between prime triplets {p, p+2, p+6}

   1st triplet:    2nd triplet:   Gap g₃(p): 
              5              11           6
             17              41          24
             41             101          60
            107             191          84
            347             461         114
            461             641         180
            881            1091         210
           1607            1871         264
           2267            2657         390
           2687            3251         564
           6197            6827         630
           6827            7877        1050
          39227           40427        1200
          46181           47711        1530
          56891           58907        2016
          83267           86111        2844
         167621          171047        3426
         375251          379007        3756
         381527          385391        3864
         549161          553097        3936
         741677          745751        4074
         805031          809141        4110
         931571          937661        6090
        2095361         2103611        8250
        2428451         2437691        9240
        4769111         4778381        9270
        4938287         4948631       10344
       12300641        12311147       10506
       12652457        12663191       10734
       13430171        13441091       10920
       14094797        14107727       12930
       18074027        18089231       15204
       29480651        29500841       20190
      107379731       107400017       20286
      138778301       138799517       21216
      156377861       156403607       25746
      242419361       242454281       34920
      913183487       913222307       38820
     1139296721      1139336111       39390
     2146630637      2146672391       41754
     2188525331      2188568351       43020
     3207540881      3207585191       44310
     3577586921      3577639421       52500
     7274246711      7274318057       71346
    33115389407     33115467521       78114
    97128744521     97128825371       80850
    99216417017     99216500057       83040
   103205810327    103205893751       83424
   133645751381    133645853711      102330
   373845384527    373845494147      109620
   412647825677    412647937127      111450 
   413307596957    413307728921      131964
  1368748574441   1368748707197      132756
  1862944563707   1862944700711      137004
  2368150202501   2368150349687      147186
  2370801522107   2370801671081      148974
  3710432509181   3710432675231      166050
  5235737405807   5235737580317      174510
  8615518909601   8615519100521      190920
 10423696470287  10423696665227      194940
 10660256412977  10660256613551      200574
 11602981439237  11602981647011      207774
 21824373608561  21824373830087      221526
 36385356561077  36385356802337      241260
 81232357111331  81232357386611      275280
186584419495421 186584419772321      276900
297164678680151 297164678975621      295470
428204300934581 428204301233081      298500
450907041535541 450907041850547      315006
464151342563471 464151342898121      334650
484860391301771 484860391645037      343266
666901733009921 666901733361947      352026

Maximal gaps between prime triplets {p, p+4, p+6}

   1st triplet:    2nd triplet:   Gap g₃(p): 
              7              13           6
             13              37          24
             37              67          30
            103             193          90
            307             457         150
            457             613         156
            613             823         210
           2137            2377         240
           2377            2683         306
           2797            3163         366
           3463            3847         384
           4783            5227         444
           5737            6547         810
           9433           10267         834
          14557           15643        1086
          24103           25303        1200
          45817           47143        1326
          52177           54493        2316
         126487          130363        3876
         317587          321817        4230
         580687          585037        4350
         715873          724117        8244
        2719663         2728543        8880
        6227563         6237013        9450
        8114857         8125543       10686
       10085623        10096573       10950
       10137493        10149277       11784
       18773137        18785953       12816
       21297553        21311107       13554
       25291363        25306867       15504
       43472497        43488073       15576
       52645423        52661677       16254
       69718147        69734653       16506
       80002627        80019223       16596
       89776327        89795773       19446
       90338953        90358897       19944
      109060027       109081543       21516
      148770907       148809247       38340
     1060162843      1060202833       39990
     1327914037      1327955593       41556
     2562574867      2562620653       45786
     2985876133      2985923323       47190
     4760009587      4760057833       48246
     5557217797      5557277653       59856
    10481744677     10481806897       62220
    19587414277     19587476563       62286
    25302582667     25302648457       65790
    30944120407     30944191387       70980
    37638900283     37638972667       72384
    49356265723     49356340387       74664
    49428907933     49428989167       81234
    70192637737     70192720303       82566
    74734558567     74734648657       90090
   111228311647    111228407113       95466
   134100150127    134100250717      100590
   195126585733    195126688957      103224
   239527477753    239527584553      106800
   415890988417    415891106857      118440
   688823669533    688823797237      127704
   906056631937    906056767327      135390
   926175746857    926175884923      138066
  1157745737047   1157745878893      141846
  1208782895053   1208783041927      146874
  2124064384483   2124064533817      149334
  2543551885573   2543552039053      153480
  4321372168453   4321372359523      191070
  6136808604343   6136808803753      199410
 18292411110217  18292411310077      199860
 19057076066317  19057076286553      220236
 21794613251773  21794613477097      225324
 35806145634613  35806145873077      238464
 75359307977293  75359308223467      246174
 89903831167897  89903831419687      251790
125428917151957 125428917432697      280740
194629563521143 194629563808363      287220
367947033766573 367947034079923      313350
376957618687747 376957619020813      333066
483633763994653 483633764339287      344634
539785800105313 539785800491887      386574

The ratio g₃(p)/log⁴p is never greater than 0.35, i.e. maximal gap sizes are less than log p times the average gap, where p is the prime at the end of the gap.