Late prime gaps

Bobby Jacobs · 2020-02-15, 21:30

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.

mart_r · 2020-02-23, 17:06

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...?

Bobby Jacobs · 2020-02-23, 21:21

Cool! I have noticed that multiples of 3 are rare as late prime gaps.

mart_r · 2020-03-08, 14:19

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.

Bobby Jacobs · 2020-04-11, 18:45

Late prime gaps are great prime gaps!

Bobby Jacobs · 2020-06-07, 23:08

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?

mart_r · 2020-06-08, 21:15

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.

2020-02-15, 21:30	#1
Bobby Jacobs May 2018 2²·53 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... 3²·71 Posts	Okay, so here's some data: Looking at conventional gaps, out of 131 data points between 22 and 2716, 67% of the powers of two appear as late gaps (4, 8, 16, 32, 64, 256) 30% of numbers of the form 2prime (6, 10, 26, 38, 46,...), and 15% of numbers of the form 2composite (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 2p 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 <= 118q into account): Code: For gaps g = kq, 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 q2^n appear as late gaps 47% of numbers of the form qp, and 26% of numbers of the form qc 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=kq 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=kq 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=kq 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 qc is not that strong. This is understandable since a factor of 3 is "taken away" from k for the gaps g=kq. (Well, I understand it in some way, but how can I explain it properly?...) 71% of numbers of the form q2^n appear as late gaps 51% of numbers of the form qp, and 42% of numbers of the form qc 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 / 22^n*

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
Prime gaps	Terence Schraut	Miscellaneous Math	10	2020-09-01 23:49
Gaps between maximal prime gaps	Bobby Jacobs	Prime Gap Searches	52	2020-08-22 15:20
Nondecreasing prime gaps	Bobby Jacobs	Prime Gap Searches	6	2019-07-12 15:26
Prime gaps above 2^64	Bobby Jacobs	Prime Gap Searches	11	2018-07-02 00:28
Happy Prime Date --- sorry for being so late	petrw1	Lounge	2	2011-05-24 06:11

2020-02-23, 21:21	#3
Bobby Jacobs May 2018 2²·53 Posts	Cool! I have noticed that multiples of 3 are rare as late prime gaps.

2020-04-11, 18:45	#5
Bobby Jacobs May 2018 2²×53 Posts	Late prime gaps are great prime gaps!

2020-06-07, 23:08	#6
Bobby Jacobs May 2018 2²·53 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?