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 2020-02-15, 21:30 #1 Bobby Jacobs     May 2018 3×79 Posts Late prime gaps A late prime gap is a prime gap of size n after a prime p such that all possible prime gaps less than n occur before p. For example, 12 is a late prime gap because the first prime gap of size 12 is between 199 and 211, and all possible prime gaps less than 12 are 1, 2, 4, 6, 8, 10, which all occur before 199. Here is the sequence of late prime gaps. 1, 2, 4, 6, 8, 10, 12, 16, 26, 28, ... This sequence is A100180 in OEIS. I wonder what patterns are in the sequence of late prime gaps.
 2020-02-23, 17:06 #2 mart_r     Dec 2008 you know...around... 2AD16 Posts Okay, so here's some data: Looking at conventional gaps, out of 131 data points between 2*2 and 2*716, 67% of the powers of two appear as late gaps (4, 8, 16, 32, 64, 256) 30% of numbers of the form 2*prime (6, 10, 26, 38, 46,...), and 15% of numbers of the form 2*composite (12, 28, 30, 36, 56,...) This is what I spoke about in the past when I said that powers of two and gaps of the form 2*p are "hardest to find". This is not too much data, so I also looked at gaps in residue classes, gaps between primes p congruent to r mod q, for q in the set {2, 4, 8, 16, 32, 64, 128, 256, 512, 1024}. This is what came up (taking gaps <= 118*q into account): Code: For gaps g = k*q, k appears # of times (out of 10) as a late gap 2 7 3 7 4 8 5 7 6 6 7 7 8 5 9 5 10 8 11 3 12 3 13 9 14 4 15 3 16 7 17 6 18 4 19 4 20 3 21 3 22 5 23 6 24 2 25 2 26 7 27 0 28 4 29 3 30 3 31 7 32 3 33 3 34 2 35 2 36 2 37 4 38 6 39 3 40 4 41 4 42 1 43 3 44 3 45 1 46 8 47 5 48 0 49 1 50 0 51 4 52 3 53 4 54 2 55 3 56 3 57 2 58 5 59 3 60 1 61 2 62 5 63 1 64 4 65 3 66 1 67 7 68 1 69 4 70 4 71 3 72 1 73 4 74 3 75 2 76 5 77 1 78 2 79 7 80 3 81 1 82 4 83 4 84 0 85 1 86 4 87 1 88 5 89 2 90 0 91 2 92 2 93 1 94 2 95 0 96 4 97 6 98 1 99 1 100 5 101 4 102 0 103 2 104 3 105 0 106 4 107 4 108 1 109 2 110 2 111 1 112 4 113 7 114 2 115 1 116 3 117 0 118 4 57% of numbers of the form q*2^n appear as late gaps 47% of numbers of the form q*p, and 26% of numbers of the form q*c This is to be interpreted as follows: Code: there are 6 powers of two up to 118, these appear 34 times in the numbers k of late gaps g=k*q for 10 values of q: 34/10/6 = 57% there are 29 primes up to 118, these appear 136 times in the numbers k of late gaps g=k*q for 10 values of q: 136/10/29 = 47% there are 82 composites up to 118, these appear 216 times in the numbers k of late gaps g=k*q for 10 values of q: 216/10/82 = 26% I also looked at q = {6, 12, 18, 24}, here the bias against gaps of the form q*c is not that strong. This is understandable since a factor of 3 is "taken away" from k for the gaps g=k*q. (Well, I understand it in some way, but how can I explain it properly?...) 71% of numbers of the form q*2^n appear as late gaps 51% of numbers of the form q*p, and 42% of numbers of the form q*c Maybe next week (or next month) I jumble up some data on twin gaps and quad gaps. Or someone else with a little more time on their hands could do it for me...? Last fiddled with by mart_r on 2020-02-23 at 17:22 Reason: mixed up 2^n / 2*2^n
 2020-02-23, 21:21 #3 Bobby Jacobs     May 2018 3·79 Posts Cool! I have noticed that multiples of 3 are rare as late prime gaps.
 2020-03-08, 14:19 #4 mart_r     Dec 2008 you know...around... 5×137 Posts So here's some data for twin prime gaps: All late gaps < 4239 (k<8e15): Code:  gap k 2 3 3 7 4 103 7 378 15 597 17 1075 19 3563 24 3843 29 6458 36 13372 43 14542 51 23277 56 25347 59 35798 64 90423 86 138187 94 213103 96 354662 99 383148 118 429182 121 614567 136 828307 144 989058 149 989443 169 1571558 171 2040992 174 2320048 189 3004313 204 4055193 216 4449232 224 4460943 228 4723290 234 6283358 235 6958413 237 8351255 239 8654928 272 9813760 276 12364972 277 12908728 289 20158288 321 34220632 361 45449332 369 51585693 386 53895292 394 54592048 409 131974248 421 142202692 446 143010737 464 165948043 479 284804898 514 386570573 561 418105097 569 461913048 574 975992038 611 1000855662 626 1526295402 666 2138760872 699 2952344623 721 2981704352 746 3024412342 766 5053512067 809 6688914953 831 7926904447 839 8054272678 864 11106595143 936 18462023822 1004 19374221148 1024 30370627668 1056 31366197567 1061 41915068062 1086 43177754272 1121 58663327587 1139 61756317343 1202 66002847468 1216 68006592497 1224 90705803338 1231 101208274662 1267 102711625448 1296 136174101612 1319 186438133443 1359 238021889388 1389 335699670478 1396 370369427962 1461 472019855132 1569 558397733113 1574 605483702178 1597 785707760883 1606 809203672447 1636 1035211477332 1644 1201386008933 1711 1437737041932 1779 2129562021213 1796 2175761168042 1824 2530550769183 1879 2910891145493 1916 3299752653517 1931 4155631964692 2001 4583730314427 2019 6221183861883 2026 7067370983472 2076 9945481356027 2236 10418063357412 2246 11286569420732 2253 12545488403432 2286 12570110501072 2301 13825779624407 2311 15184235238237 2321 15516154655937 2361 19323545584812 2416 20104392797057 2430 20317390689250 2446 33847946793892 2456 39789986883582 2544 43734549646928 2599 48528321238833 2631 61546648657772 2694 61693448191183 2716 62848316218142 2722 88837450586533 2729 90991395906108 2754 97180557658683 2819 107795823415758 2866 142935886049397 2949 175244695686518 2964 203347509247523 3004 206053016592208 3006 209662169007197 3016 264095286829287 3104 267370390859663 3111 316152757567642 3144 399795784786828 3154 433648726364318 3244 491668861876693 3314 509150732932538 3321 612211013028367 3341 684725355860402 3396 956548812672742 3449 1065846223264498 3559 1415931820757328 3614 1545634951552158 3659 1689960282469393 3671 1750926770970442 3676 2032554433644717 3779 2063415047832358 3809 2075503685369503 3847 2146254663929243 3849 2266903514297923 3902 2286066324394743 3904 2391528479271188 3919 2665065580231183 3924 3226495075272073 3959 3226565924675093 3991 3324584193835277 3994 4936819769069403 4014 5908912848885783 4161 6198884291928232 4206 6650801286755762 4239 >8e15 Number of gaps between k's where k corresponds to a twin prime 6*k±1 Code: gap of which mod 5 1001 2001 3001 <4239 <1000 -2000 -3000 -4000 0 3 2 0 1 0 1 66 24 15 16 9 2 12 6 3 1 2 3 5 4 0 1 0 4 72 30 13 9 18 total late % comp. 3658 125 3.42% prime 580 33 5.69% Regarding the gaps between k's where k corresponds to a prime quadruplet 30*k+[11,13,17,19], only five of the late gaps are not 1 or 6 mod 7. This is nicely illustrated in the attached graphs. All late gaps < 142003: Code:  1 1006301 4 1022381 6 3512051 20 12390011 41 181773281 216 258578051 477 449686421 771 483751781 776 501949571 939 901797101 972 2280695771 1842 3318979421 2633 4443215471 3184 4519480571 3205 5272815671 3634 5273110691 3877 5727501581 4451 6472241381 4495 12950441681 5265 15998125061 5669 18110108111 5825 22736410391 6672 26337289631 7076 29431834121 7179 30364737041 7279 32496998111 7797 43206363911 8784 60483913151 9395 92840696951 9442 126228731801 12237 139083671561 12678 155878166831 13094 181390479371 13614 185647034381 13887 313724920121 15968 590556766361 19004 650264532551 20126 673858295441 20420 897526840751 21764 1206449526011 22700 1328913667841 24100 1563904181291 25964 2280895284131 27418 2289721396421 28211 2448682662911 29091 2715434952941 29576 2915601088241 29751 3238232397731 30444 3698188741781 32388 3844976186531 32644 4627567062191 34082 5512505298731 36007 7934854558061 37428 9266276470301 38900 10513541415071 40272 10822965113921 40363 13454097713261 41315 13561209164141 43688 14451951842351 44120 15347392702121 44820 18749825413211 46271 19817962791461 46353 21244300087091 46628 25121629766501 47018 27644008633931 48539 28192286706761 49176 28631003371511 52048 37365358179551 53836 53562450491981 55721 55042761501191 61221 69634261830641 61986 75778627165061 62378 80642431548821 65022 90399894584801 66492 93019854565901 66634 109692595765391 66851 113930607883451 68545 115430448968501 68669 144851526543791 71751 160784975572601 73641 166963851729131 76980 169784862027041 77477 218580135884651 80851 228379187756771 81649 271291235724791 81712 336343866603131 84517 344941940619581 87137 367507999705961 87506 381555427314401 88591 453794043872201 92756 465101997604031 93493 470405616730331 94032 471744455264651 94898 533909739050321 95957 631088856952451 97915 815257366441361 101305 867486371239721 101599 882066274243961 104231 961361063547341 106135 1024650550234691 108949 1098480432385751 109717 1138879053672791 110928 1201344536919881 113728 1435753079466941 115627 1494573191034611 116640 1613830400112641 118686 1666705515427871 119013 2001277766145011 121248 2006231330147501 122690 2373240553817051 127359 2764746940114301 131412 2780980757625611 131468 3009352033185521 134149 3433340976250061 134975 3459652600956701 136660 3638214934057001 138732 3648999618996491 139278 4610874166974191 141189 5066604752118851 142003 >5.1E+15 total late % comp. 128819 99 0.077% prime 13184 21 0.159% BTW, @Robert: you might want to check your list for the gap entries g=2077 and 11076. Attached Thumbnails
 2020-04-11, 18:45 #5 Bobby Jacobs     May 2018 3·79 Posts Late prime gaps are great prime gaps!
 2020-06-07, 23:08 #6 Bobby Jacobs     May 2018 ED16 Posts Here are the known gaps that are both maximal prime gaps and late prime gaps. 1, 2, 4, 6, 8, 36 Are there any more?
2020-06-08, 21:15   #7
mart_r

Dec 2008
you know...around...

5·137 Posts

Quote:
 Originally Posted by Bobby Jacobs Here are the known gaps that are both maximal prime gaps and late prime gaps. 1, 2, 4, 6, 8, 36
*Sigh* Okay, I have a go at it. But only because I have some exclusive data.
Code:
q: k for which there exist maximal gaps q*k after a prime p in an arithmetic progression p+q*k where all smaller first occurrence gaps have smaller initial primes
2: 1, 2, 3, 4, 18
4: 1, 2, 3, 4, 5, 6, 16, 17
6: 1, 2, 3, 6, 7, 10, 15
8: 1, 2, 3, 4, 5, 8, 39, 40
10: 1, 2, 3, 9, 10, 11, 12, 13
12: 1, 2, 3, 4, 5, 15
14: 1, 4, 5, 6, 9, 10, 11, 12, 31, 32
16: 1, 2, 6, 7, 22
18: 1, 2, 3, 4, 5, 6, 9, 10, 11, 126
20: 1, 2, 3, 4, 24
22: 6, 7, 14, 15
24: 1, 2, 3, 4, 5, 6, 7, 12, 36
26: 1, 2, 3, 6, 7, 8
28: 1, 2, 3, 4, 5, 6, 10, 11
30: 1, 2, 3, 4, 7, 10, 18, 28
32: 3, 4, 7, 12
34: 1, 2, 12, 13, 51
36: 1, 7, 8, 9, 10, 19, 20
38: 1, 2, 8, 15
40: 1, 9, 20, 21
42: 1, 2, 3, 4, 5, 6, 7, 8, 31, 35, 40, 41
44: 1
46:
48: 1, 2, 3, 6, 7, 16
50: 1, 2, 6, 7, 12, 27
52: 3
54: 1, 4, 5, 6, 13, 14, 15, 20
56: 1
58: 1, 4, 7
60: 1, 2, 3, 4, 5, 8, 12, 39
62:
64: 1, 4, 5, 6
66: 1, 2, 3, 4, 11, 17, 27
68: 1, 4, 5, 6
70: 1, 2, 3, 6, 50, 67
72: 5, 11, 12, 13, 14, 15, 25, 26, 29, 30
74: 3, 9, 28
76: 1, 2
78: 1, 2, 3, 4, 8, 21, 22
80: 1, 2, 3, 6, 14, 38, 39
82: 8
84: 1, 2, 5, 17, 21, 22, 23, 24
86: 1, 4, 5, 6, 7, 19
88: 4
90: 1, 7, 8, 13, 27, 31, 48
92:
94: 1, 2
96: 1, 2, 3, 4, 12, 13
98: 1, 4, 5, 6, 7
100: 1, 2, 3, 4, 5, 6, 28
102: 1, 5
104: 1, 10, 14, 15
106: 1, 13
108: 1, 2, 14, 21, 53
110: 1, 2, 3, 4, 5, 6, 18, 31
112: 10, 11, 12, 44, 45
114: 6, 9, 22, 47
116: 5, 12, 30
118:
120: 1, 4, 5, 6, 7, 15, 32, 33
122: 13
124: 1, 9
126: 1, 2, 3, 4, 5, 11, 16
128: 1, 30, 31
130: 3, 4, 5, 6, 7, 25
132: 1, 2, 3, 4, 7, 8, 21, 22, 32, 33, 34, 35, 106
134: 1, 15, 25
136: 1, 2, 7
138: 3, 4, 5, 8
140: 3
142:
144: 1, 7, 16
146: 1, 26
148: 1, 5, 9, 29
150: 1, 2, 5, 8, 9, 13
152: 9, 14
154: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15
156: 5, 6, 11, 26, 58
158: 8, 20, 21
160: 1, 2, 3, 15
162: 1, 2, 7, 15, 25
164: 1, 8, 9
166: 26, 27
168: 1, 2, 3, 6, 7, 8, 11, 12, 44, 45
170: 1, 2, 3, 19, 20, 73
172:
174: 1, 2, 5, 6, 9, 20, 34, 35
176: 1, 2, 3, 6, 7, 21
178: 1
180: 3, 4, 5, 6, 17, 23, 26, 27, 28, 29, 30, 40, 49, 50
182: 5
184: 8, 12, 13
186: 1, 2, 3, 6, 9, 10, 11, 12, 15, 16
188: 1, 2
190: 1, 9, 12, 27, 28, 44
192: 1, 4, 5, 8, 11
194: 1, 12, 30
196: 1, 2, 3
198: 4, 5, 6, 7, 8, 14, 22, 25, 40, 41
200: 11, 12, 15, 16, 20
202:
204: 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 18
206:
208: 1, 2, 3, 10, 11, 12, 15
210: 3, 4, 5, 8, 9, 10, 11, 12, 13, 46, 47
212: 12, 15
214: 5
216: 5, 6, 7, 15, 23, 44
218:
220: 1, 2, 7, 8, 9, 18
222: 1, 2, 7, 25, 33, 45
224: 1, 4, 5, 6, 7
226: 1, 2, 10, 11, 16, 26, 27
228: 1, 2, 3, 11
230: 1, 2, 6, 19
232: 7, 8, 9, 22
234: 1, 2, 5, 15
236: 1, 2, 3
238: 1, 55
240: 3, 7, 8, 9, 21, 42, 51
242:
244: 25
246: 1, 2, 3, 9, 10, 11
248: 1
250: 3, 9, 10
252: 1, 5, 6, 7, 11, 24
254: 1, 5, 13, 14
256:
258: 1, 2, 8, 9, 10, 11, 12
260: 1, 139
262:
264: 1, 16, 46, 51
266: 1, 6, 7, 8
268: 1, 2, 3, 51
270: 1, 2, 3, 4, 5, 6, 7, 11, 18, 19, 20
272: 3, 4
274: 1
276: 1, 2, 5, 8, 9, 10, 11, 25
278: 1, 2
280: 1, 2, 3, 6, 14, 15, 16, 29
282: 3, 10, 22, 32
284: 6, 7, 12
286: 6, 7, 18, 21
288: 1, 18
290: 1, 2, 3, 6, 14
292:
294: 7, 12
296: 41
298:
300: 1, 2, 5, 6, 7, 8, 9, 10, 11
302: 49
304: 1, 2, 3, 12, 38
306: 1, 5, 10, 58
308: 1, 14, 15
310: 1, 2, 7
312: 1, 2, 3, 4, 7, 8, 9, 21
314: 1
316:
318: 3, 4, 5, 6, 7, 8, 11
320: 3, 4, 5, 6, 9, 13, 14
322: 30, 31, 78
324: 6, 7
326: 27
328: 1, 2, 3
330: 1, 2, 5, 6, 11, 12, 15, 16, 17, 18, 31
332:
334: 1, 2, 3
336: 3, 9, 10, 11, 12
338: 13, 14
340: 10
342: 1, 8, 9, 10, 11, 12, 13, 14
344: 1
346: 1, 8, 9
348: 1, 4, 16, 20
350: 1
352:
354: 1, 4, 5, 6, 7, 30, 50, 53
356: 1
358: 16, 24
360: 1, 4, 5, 6, 7, 14, 15
362: 3
364: 1, 2, 3, 4, 5, 30
366: 4, 5, 8
368: 3, 4, 5, 6
370: 1, 97
372: 30, 36, 37, 38
374: 3, 6, 10, 39
376: 1, 2
378: 1, 2, 3, 36, 48, 49
380: 1
382:
384: 1, 5
386: 1
388: 10, 11
390: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
392: 3, 13, 19, 32
394: 1, 7, 8
396: 1, 6, 7, 8, 13, 14, 21
398: 1, 14
400: 6, 33
402: 3, 4, 7, 8, 9, 10
404: 13
406: 1, 41, 42
408: 7, 15
410: 18
412:
414: 1, 2, 5, 6
416: 1, 2, 3
418: 1, 27
420: 1, 2, 3, 4, 8, 11, 23, 74
422:
424: 5
426: 1, 2, 3, 10, 51, 62
428: 1, 2, 3, 4
430: 1, 9, 10, 93, 94
432: 24, 34, 126
434:
436: 1, 2
438: 1, 2, 3, 10, 16
440: 1, 2, 15, 16
442:
444: 1, 5, 6
446: 1, 6, 7
448: 9
450: 1, 2, 3, 6, 7, 13, 28
452:
454: 1, 6, 7
456: 1, 4, 10, 11, 12, 13
458: 1
460: 1, 12, 13
462: 1, 2, 6, 9, 10, 19, 20, 21, 28
464: 1, 8
466: 5
468: 5, 6, 7, 11, 14, 20, 21, 29, 30, 31, 32, 33
470: 11, 12
472:
474: 1, 14
476: 1, 6
478:
480: 1, 9, 10, 11, 12, 13, 17, 18, 23, 24, 43
482:
484: 1
486: 1, 17, 21, 22, 23, 24, 25
488: 1, 10
490: 3, 4, 17
492: 4
494: 3, 4, 15
496: 1, 2, 3, 4, 21, 40, 41
498: 1, 5, 20, 117
500: 1, 13
502:
504: 1, 2, 17, 18
506: 1, 4, 27
508: 6, 36
510: 6, 7, 8, 9, 10, 11, 20, 27
512:
514: 26, 27
516: 1, 6, 9, 10, 11
518: 1, 5, 6, 15
520: 1, 6, 7, 105
522: 5, 9, 10, 15
524:
526:
528: 3, 4, 5, 18, 32
530: 13
532: 11, 12, 13, 14, 15, 18, 22, 100
534: 48
536: 27
538: 1, 15, 24
540: 1, 2, 6, 7, 10, 23, 24, 25, 115
542: 3
544: 1, 2
546: 3, 7, 14, 15, 16, 17, 66
548: 34
550: 15, 16
552: 1, 4, 13, 21, 22, 30, 57
554: 1
556: 17, 18
558: 1, 2, 10, 13, 16
560: 1, 14, 15, 19, 20, 21, 24
562: 15
564: 1, 21, 25
566: 1, 6, 7, 14, 28
568: 1
570: 1, 2, 9, 10, 11, 12, 13, 18, 25
572: 15
574: 1, 2, 3
576: 11
578: 28, 39
580: 15
582: 1, 2, 3
584: 1, 5, 6
586: 20, 21
588: 1, 2, 3, 26
590: 1, 2, 3, 10, 33, 34
592: 24, 25
594: 1, 22
596: 1
598: 1, 2, 3, 15
600: 1, 8, 14, 38, 39, 40, 64
602: 12
604: 1, 2, 5, 6
606: 5, 6, 7, 33, 53
608:
610: 1, 2, 29
612: 1, 6, 7, 25, 26
614: 1
616: 1, 2, 3, 7
618: 11, 23, 24
620:
622: 24, 25
624: 4, 5, 6
626: 10, 15, 47
628: 1
630: 1, 2, 7, 8, 9, 10, 11, 12, 22, 23, 24, 25, 50, 58, 75
632:
634: 5, 50
636: 1, 9, 10, 11, 16, 34, 174
638: 1, 2, 3, 4, 36, 37
640: 1, 2, 3
642: 1, 2
644: 1
646: 64
648: 1, 2
650: 1, 2, 7
652:
654: 1, 2, 3, 4, 5, 10, 18, 19, 32, 33
656: 1, 2, 3
658: 1, 2, 3, 6, 7, 8
660: 3, 13, 14, 24, 91
662: 23
664: 43
666: 5, 13, 16, 56
668:
670: 1
672: 1, 4, 9, 10, 11, 12, 13, 34, 43, 44
674: 1, 23
676:
678: 1, 5, 6, 7
680: 1, 2
682:
684: 3, 7, 8, 9, 37, 43
686: 7
688: 1, 2
690: 5, 10, 11, 43
692:
694: 10
696: 1, 2, 3, 4, 23
698: 1
700: 9, 10
702: 8, 9, 16, 17
704: 6
706: 1
708: 6, 7, 15, 16, 17
710: 8, 15, 16, 23
712:
714: 1, 2, 6, 7, 8, 18, 36, 37, 40
716: 1
718:
720: 1, 2, 5, 6, 10, 11, 53
722: 3
724: 1, 2, 3, 4, 5, 6
726: 5, 13
728: 42
730: 1, 5, 6, 21
732:
734:
736: 1, 4, 21
738: 1, 2, 3
740: 1, 2, 3, 4, 5, 6
742:
744: 3, 4, 5, 8, 9, 10
746: 6, 18
748: 1, 34
750: 1, 18, 22, 42
752:
754: 1, 17
756: 1, 2, 12, 13, 37, 48
758: 1, 5
760:
762: 30
764: 11, 30
766: 1
768: 1, 2, 5, 6, 7, 8, 9, 14, 22, 27
770: 1, 2, 3, 9, 10, 11, 12, 13, 14, 15, 16, 27, 34
772:
774: 29, 30, 31
776: 36
778: 23, 24
780: 1, 2, 3, 4, 5, 10, 11, 15, 42, 43
782: 3, 42
784: 1
786: 20
788: 21
790: 3, 4, 7, 8
792: 1
794: 1, 57
796: 5, 16, 17
798: 5, 6, 11, 15, 16, 22, 23, 24
800: 20, 30, 31
802: 3, 4
804: 1, 2, 6, 7, 10, 11, 12, 13, 18
806: 1, 2
808: 1, 2, 26, 27
810: 27
812: 6, 19, 20, 21
814: 7, 8, 9, 10, 34
816: 1, 6, 7, 8, 9, 23, 24
818: 1, 19
820: 1, 7, 8, 15, 22, 23, 24, 25
822: 1, 6, 14, 15
824: 1, 10, 13
826: 1, 2, 6, 7, 8, 9, 16, 81, 82
828: 9, 10, 18
830: 18
832:
834: 1, 10, 23
836: 1
838: 11, 12
840: 7, 8, 9, 13, 65
842: 6, 7
844:
846: 3, 4, 5
848: 3
850: 1, 17, 18, 23, 24
852: 1, 11, 31
854: 1, 16, 23, 39
856: 1, 29
858: 1, 2, 8, 9, 10, 11, 20, 21, 22
860: 1, 12, 27
862: 26, 27, 28
864: 5, 27
866:
868:
870: 1, 2, 3, 4, 8, 14, 51
872:
874: 1, 2
876: 1, 6, 7, 8, 14, 15, 16
878: 1
880: 1, 5, 52, 53, 54
882: 1, 4, 5, 6, 12
884: 1
886: 40
888: 6, 7, 30
890: 5, 6, 23, 51
892: 22
894: 8, 47
896: 22, 23, 24
898:
900: 1, 15, 16, 22, 38
902: 6, 7
904: 1
906: 1, 18
908: 1, 22
910: 15
912: 12, 29, 32, 33, 34
914: 3
916: 1, 6, 7, 25
918: 14, 15, 16
920:
922: 16, 17
924: 1, 2, 5, 13
926: 1, 16
928: 27, 28
930: 1, 2, 3, 4, 10, 13, 16, 17, 18
932: 11, 12
934: 1, 2, 3
936: 1, 2
938: 1, 9, 10, 11, 30
940: 23
942: 1
944: 1, 20
946:
948: 1, 4, 5
950: 1, 2, 3, 4, 5
952: 20, 21
954: 8
956: 33
958:
960: 1, 2, 11, 22, 23
962: 14
964: 1, 2, 3
966: 1, 10, 11
968: 1
970: 19
972: 1, 2, 11, 17, 18, 31
974: 1
976:
978: 1, 11, 14, 28, 54
980: 1, 2, 3, 4, 5, 6, 7, 13, 14
982:
984: 3, 6, 7, 14
986:
988: 1, 6, 23
990: 1, 2, 3, 6, 7, 10, 15, 16, 25
992:
994: 1, 2, 7, 19, 20, 21, 22
996: 5, 6, 10, 68, 69
998:
1000: 18

max.: 174 @ q=636
avg. max.: 23.56

stats:
k count
1 276
2 134
3 113
4 79
5 94
6 104
7 87
8 60
9 57
10 65
11 59
12 46
13 41
14 37
15 50
16 34
17 19
18 28
19 13
20 22
21 26
22 24
23 25
24 21
25 19
26 11
27 21
28 12
29 9
30 17
31 12
32 9
33 10
34 11
35 3
36 7
37 5
38 5
39 7
40 8
41 5
42 6
43 6
44 5
45 3
46 2
47 4
48 4
49 3
50 5
51 7
52 1
53 5
54 2
55 1
56 1
57 2
58 3
62 1
64 2
65 1
66 1
67 1
68 1
69 1
73 1
74 1
75 1
78 1
81 1
82 1
91 1
93 1
94 1
97 1
100 1
105 1
106 1
115 1
117 1
126 2
139 1
174 1
The k's for q=2 correspond to the cases for classical prime gaps, i.e. the ones you mentioned, only half in size.

The list should be finite for each q with probability one. A proof thereof is left as an exercise to the experts.

Quote:
 Originally Posted by Bobby Jacobs Are there any more?
Fill in the 31 unknown first occurrence gaps between 1432 and 1548 without finding a gap bigger than 1550, and hope that the subsequent CFC is exactly 1552 in size.
In other words: certainly not.

A really hard problem would be to determine whether or not there is a global maximum for k in the list above for lim q --> inf.

Last fiddled with by mart_r on 2020-06-08 at 21:59 Reason: k=1 by default was not quite correct

2021-08-11, 21:30   #8
Bobby Jacobs

May 2018

3·79 Posts

Quote:
 Originally Posted by mart_r Fill in the 31 unknown first occurrence gaps between 1432 and 1548 without finding a gap bigger than 1550, and hope that the subsequent CFC is exactly 1552 in size.
We did not fill in any of the unknown first occurrence gaps under 1550, but the next maximal prime gap is 1552.

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