Ed Pegg Jr wrote:

> 5 is needed for a primegap of 6,
> 7 is needed for a primegap of 22,
> 11 is needed for a primegap of 116, and so on.

g is the smallest gap which has the first occurrence from p to p+g.
Below are results from an exhaustive search for g < 3*10^10.

p g
---- -----------
3 2
5 6
7 22
13 88
19 112
11 116
31 202
29 242
23 284
17 470
43 718
37 772
41 1326
53 1328
47 1334
67 1642
79 1732
61 1762
71 2402
59 2558
73 3274
101 5246
149 5888
83 7094
127 7702
97 7984
89 9512
109 9952
137 9974
157 10342
107 10532
103 12688
151 13528
113 16766
163 25678
139 25708
181 37666
197 59894
131 60458
193 61756
167 62156
191 69500
173 69518
227 76004
199 76168
179 76838
223 81784
211 100558
307 103102
241 103528
349 108412
229 108532
239 111512
233 139058
257 162134
379 180814
277 190006
271 190096
373 194218
263 250280
283 286114
269 294404
409 359662
251 365258
331 406918
337 413242
313 416704
281 430418
317 438314
311 460232
347 484652
293 610124
431 651362
359 676412
421 677758
367 690634
419 713618
397 774376
619 872698
389 1017902
467 1053116
353 1053848
401 1067816
433 1229638
439 1334848
383 1385444
503 1514984
449 1568648
487 1911352
479 2017034
547 2282872
443 2346404
523 2359474
631 2373478
457 2491042
463 2498458
571 2738728
461 2740622
541 2902342
509 3132044
521 3591806
499 4005292
593 4874258
617 4910546
569 5081192
491 5768912
563 5832878
661 6088792
557 7099472
691 7642528
607 7655806
643 8042218
601 8734948
641 8914196
613 9012928
751 9244552
727 9878542
577 11991484
757 13038352
811 13591192
587 13720004
719 14808734
787 15412972
709 16708792
673 16766338
647 17211872
769 18063208
599 19171058
739 19700218
773 22176548
733 22671958
677 22842824
829 24318988
653 27349526
701 28160402
683 31480166
797 32602814
821 34726238
1093 34857778
853 37572784
883 39127618
877 40203172
743 41570636
839 42018188
857 44688542
659 47225708
991 48726448
919 50361532
823 53936758
859 54545692
1117 55076404
953 55115978
827 56808482
907 61334716
761 65396288
929 73094972
937 73410502
1033 75884164
881 76097852
947 79296956
809 86388902
967 96330292
997 109611364
1051 112930942
1087 125676016
863 131079974
887 137987162
1217 147838742
977 154861334
911 164594336
1039 172682308
1021 173260042
1009 178260934
1069 185472202
1103 238205144
1031 239201918
1109 251531444
1279 255172912
1129 268571944
1019 319732832
1123 336671788
971 360236108
1013 374378834
1237 382362826
1181 398579282
1229 402377288
1091 435146828
1277 441358094
1171 441573238
941 444571916
1063 451714258
1153 474819178
1061 476723492
1423 532932016
1249 536796808
1049 541913234
1097 551437862
983 566051996
1223 593888264
1231 610710082
1201 620215306
1381 636335578
1291 645446962
1373 647722436
1187 648067634
1409 658548458
1213 673805014
1303 735935908
1321 785148508
1151 824722442
1453 847434106
1481 916861118
1429 935409214
1297 991647586
1301 1028680076
1307 1154038124
1163 1196773574
1289 1257220178
1259 1268850092
1193 1392085826
1367 1602623234
1531 1606274038
1361 1746162182
1549 1834890388
1451 1873545218
1327 2010159922
1399 2058479002
1579 2136557938
1511 2177804666
1283 2271372674
1471 2544495082
1459 2575289824
1759 2583022822
1319 2908496564
1427 3026372642
1877 3220560764
1597 3223205122
1609 3270099058
1483 3287328298
1489 3478603012
1699 3623443228
1433 3652026998
1447 3916848772
1499 4060828922
1567 4227211162
1487 4300211324
1439 4717151294
1627 4738962244
1607 4796950934
1559 5121642404
1583 5122716218
2239 5880449914
1543 5942175436
1553 6530792714
1789 6762499438
1783 7260663988
1493 7288310924
1777 7364581984
1663 7459545466
1571 7507539476
1753 7732777354
1723 7975223296
1879 8359726222
1523 8988682988
1669 9025905478
1637 9059125532
1621 9103725262
1657 9249017152
1747 9274383442
1619 9627720644
1873 9709006654
1693 10009858036
1697 10133678774
1601 10605247472
1787 10790071076
1741 10841132716
1831 11529890332
2141 11691729188
1667 11810687576
1811 13374986246
1823 13617424424
1801 14551617628
1931 14999483288
1871 15731703242
2083 16964487538
1613 17190939806
1847 17989476584
2063 18265616426
1709 19270456922
1993 19553504956
1861 19705555288
2143 20405452048
1867 20440585822
1933 22279195054
1721 22410653816
1997 23023103954
2113 23107401658
1901 25156426202
1733 25473995978
1951 27783214582

The two last "champions" (larger p than any smaller g) are:
1877 3220560764
2239 5880449914

5880449914 requires 45 more primes than the previous champion.
A surprisingly large jump, still unbeaten at 3*10^10.

--
Jens Kruse Andersen