Maximal gaps between prime quintuplets

Prime quintuplets (5-tuples) are the densest permissible clusters of 5 consecutive primes. There are two types of prime quintuplets:

{p, p+2, p+6, p+8, p+12} (OEIS A022006, A201073, A201074, A233432)

{p, p+4, p+6, p+10, p+12} (OEIS A022007, A201062, A201063, A233433).

The observed maximal gaps between prime quintuplets 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 g₅(p)) = a(log(p/a) − 2/5) = O(log⁶p)

where a = 0.098699 log⁵p is the average gap, as predicted by the Hardy-Littlewood k-tuple conjecture.

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

Maximal gaps between prime quintuplets {p, p+2, p+6, p+8, p+12}

1st quintuplet:   2nd quintuplet:   Gap g₅(p): 
              5               11             6
             11              101            90
            101             1481          1380
           1481            16061         14580
          22271            43781         21510
          55331           144161         88830
         536441           633461         97020
         661091           768191        107100
        1461401          1573541        112140
        1615841          1917731        301890
        5527001          5928821        401820
       11086841         11664551        577710
       35240321         35930171        689850
       53266391         54112601        846210
       72610121         73467131        857010
       92202821         93188981        986160
      117458981        119114111       1655130
      196091171        198126911       2035740
      636118781        638385101       2266320
      975348161        977815451       2467290
     1156096301       1158711011       2614710
     1277816921       1281122231       3305310
     1347962381       1351492601       3530220
     2195593481       2199473531       3880050
     3128295551       3132180971       3885420
     4015046591       4020337031       5290440
     8280668651       8286382451       5713800
     9027127091       9033176981       6049890
    15686967971      15693096311       6128340
    18901038971      18908988011       7949040
    21785624291      21793595561       7971270
    30310287431      30321057581      10770150
   107604759671     107616100511      11340840
   140760439991     140772689501      12249510
   162661360481     162673773671      12413190
   187735329491     187749510491      14181000
   327978626531     327994719461      16092930
   508259311991     508275672341      16360350
   620537349191     620554105931      16756740
   667672901711     667689883031      16981320
  1079628551621    1079646141851      17590230
  1104604933841    1104624218981      19285140
  1182148717481    1182168243071      19525590
  1197151034531    1197173264711      22230180
  2286697462781    2286720012251      22549470
  2435950632251    2435980618781      29986530
  3276773115431    3276805283951      32168520
  5229301162991    5229337555061      36392070
  9196865051651    9196903746881      38695230
 14660925945221   14660966101421      40156200
 21006417451961   21006458070461      40618500
 22175175736991   22175216733491      40996500
 22726966063091   22727007515411      41452320
 22931291089451   22931338667591      47578140
 31060723328351   31060771959221      48630870
 85489258071311   85489313115881      55044570
 90913430825291   90913489290971      58465680
 96730325054171   96730390102391      65048220
199672700175071  199672765913051      65737980
275444947505591  275445018294491      70788900
331992774272981  331992848243801      73970820
465968834865971  465968914851101      79985130
686535413263871  686535495684161      82420290
761914822198961  761914910291531      88092570

Maximal gaps between prime quintuplets {p, p+4, p+6, p+10, p+12}

1st quintuplet:   2nd quintuplet:   Gap g₅(p): 
              7               97            90
             97             1867          1770
           3457             5647          2190 
           5647            15727         10080
          19417            43777         24360
          43777            79687         35910
         101107           257857        156750
        1621717          1830337        208620
        3690517          3995437        304920
        5425747          5732137        306390
        8799607          9127627        328020
        9511417          9933607        422190
       16388917         16915267        526350
       22678417         23317747        639330
       31875577         32582437        706860
       37162117         38028577        866460
       64210117         65240887       1030770
      119732017        120843637       1111620
      200271517        201418957       1147440
      203169007        204320107       1151100
      241307107        242754637       1447530
      342235627        344005297       1769670
      367358347        369151417       1793070
      378200227        380224837       2024610
      438140947        440461117       2320170
      446609407        448944487       2335080
      711616897        714020467       2403570
      966813007        970371037       3558030
     2044014607       2048210107       4195500
     3510456787       3514919917       4463130
     4700738167       4705340527       4602360
     5798359657       5803569847       5210190
     7896734467       7902065527       5331060
    12654304207      12659672737       5368530
    13890542377      13896088897       5546520
    14662830817      14668797037       5966220
    15434185927      15440743597       6557670
    17375054227      17381644867       6590640
    17537596327      17544955777       7359450
    25988605537      25997279377       8673840
    66407160637      66416495137       9334500
    74862035617      74871605947       9570330
    77710388047      77723371717      12983670
   144124106167     144138703987      14597820
   210222262087     210238658797      16396710
   585234882097     585252521167      17639070
   926017532047     926036335117      18803070
   986089952917     986113345747      23392830
  2819808136417    2819832258697      24122280
  3013422626107    3013449379477      26753370
  3538026326827    3538053196957      26870130
  4674635167747    4674662545867      27378120
  5757142722757    5757171559957      28837200
  7464931087717    7464961813867      30726150
  8402871269197    8402904566467      33297270
  9292699799017    9292733288557      33489540
 10985205390997   10985239010737      33619740
 12992848206847   12992884792957      36586110 
 15589051692667   15589094176627      42483960 
 24096376903597   24096421071127      44167530 
 37371241083097   37371285854467      44771370 
 38728669335607   38728728308527      58972920 
 91572717670537   91572784840627      67170090 
109950817237357  109950886775827      69538470 
325554440818297  325554513360487      72542190 
481567288596127  481567361629087      73032960 
501796510663237  501796584764467      74101230 
535243109721577  535243185965557      76243980 
657351798174427  657351876771637      78597210 
818872754682547  818872840949077      86266530 
991851356676277  991851464273767     107597490

The ratio g₅(p)/log⁶p is never greater than 0.0987, 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.