Gaps between non-consecutive primes

[robert44444uk]

The closest I have gotten to 100 primes following 1571162669*193#+129568114146274965711541776666046371290799466131684641935400586161726498035577 is 95 primes, within a period of 8346 compared to the well-known gap of 8350 following 29370323406802259015...95728858676728143227

sa I have devoted far too many resources to this, I will rest.

I also look briefly at the gap following 266190823030249*1129#/210-22844, but the length of time taken to check each possible range of 43k+ is too long. The best I achieved to date is 84 primes following

1101306855*1151#+67995358713657430359048762006542336703972224978670437437482633858004501532345946577534465437727848195399060224576423535081766982746433158823827486255141146637104093921266819644253660410020299599441986875748296750154110874438401578094603567430369998521465621565610168020569114152417095857527450304064588327045566434613143149884391737286419623885764232620049541559250548525133540166835094146124824189204240031275094620798491331644219231576586550944407818428480069934923985835440814277.

I found two other multipliers 1101311064 and 1101330536 giving the same 84 prime result. The closeness of the multipliers suggests that 100 primes is quite possible.

[mart_r]

Data for maximal gaps for p < 3*10¹³ and k <= 109 is now publicly available! Rejoice!
I'm probably taking this up to p = 10¹⁴. Well, unless anyone wants to join in.


Since the primes at the start of a maximal gap almost always* come in clusters, I did a quick check which p_n had the highest number of occurrences for k <= 100, for 3*10¹³ downwards:
* I know that may be a rather daring statement...

#occ   p_n
2      29418557625949  (k = 11, 16)
4      29418557625841  (k = 13, 14, 17, 18)
21     29077945916363  (55 <= k <= 85)
23     1376589410333   (55 <= k <= 87)
30     16025473729     (52 <= k <= 98)
33     3099587         (48 <= k <= 100)
34     18313           (47 <= k <= 95)
39     1621            (24 <= k <= 96)
45     661             (18 <= k <= 100)
52     467             (9 <= k <= 99)
66     283             (6 <= k <= 100)
68     199             (2 <= k <= 96)
73     109             (2 <= k <= 100)
77     7
100    2

2 and 3 always occur as primes preceding maximal gaps. 5 doesn't always occur since for p = 3 (technically p₂ = 3), for some k, p_2+k and p_2+k+1 are twin primes and in that case for p = 5 the gap length is the same as for p = 3. However, whenever 5 doesn't appear as a maximal gap, then 7 definitely does, and with respect to the number of occurrences, 7 is either in the lead by one or ties with 5. No p > 7 appears more often than p = 7 as a prime preceding a maximal gap for k = 1, 2, 3, ..., so p = 7 is a local maximum here.

But let's do this more formally:

Let \(p_n\) be the set of prime numbers and \(o_n(x)\) the set of the number of occurrences of \(p_n\) as primes preceding a maximal gap for all positive integers \(k <= x\).
\(p_n = \{2, 3, 5, 7, 11, ...\}\)
\(o_n(1) = \{1, 1, 0, 1, 0, 0, 0, 0, 1, 0, ...\}\)
\(o_n(1000) = \{1000, 1000, 827, 828, 658, 781, 660, 783, 661, 416, ...\}\)

\(o_n\) and the corresponding \(p_n\) constitutes a local maximum for the above table - in this case for x = 100 - if there does not exist \(m > n\) such that \(o_m(x) > o_n(x)\).

Conjecture: as \(k \to \infty\), the smallest \(p_n\) in the above table with a local maximum of number of occurrences as maximal gap commencers will be fixed. 19 chimes in for a larger range of \(k\), so the list of local maxima \(p_n\) will probably start {2, 7, 19, 109, 199, 9439 (?), ...} for k sufficiently large - this appears to be very tricky, at least numerically...

A follow-up question will be: for fixed x, at what point will the list of local maxima p_n be settled? For example, in the above table for x = 100, could there be a larger p_n preceding a maximal gap for more than half of the values of k (in which case o_n = 45 / p_n = 661 and possibly o_n = 52 / p_n = 467 will be superseded)? Or could there be a gap between consecutive primes so large that all - or at least most - of the p_n for k > 1 also turn out as maximal gaps?

Once creativity strikes... k = 6 is the first k for which p_n = 2, 3, 5, and 7 each start a maximal gap. For k = 12, all of the first five primes appear in the attached list. For k = 19, this makes six primes, and the first 13 (!) primes appear at k = 68 (so p_n+68-p_n becomes continually larger for every p_n <= 41). I bet MattcAnderson would like to see this sequence in the OEIS

I guess I'm biting off more than I can chew...

[mart_r]

Originally Posted by Bobby Jacobs

For each k, what are the first few gaps with record CSG ratio? This is very interesting.

These are the current record CSG for each k @ p <= 3.9*10¹³:

k   gap   CSG           p
1   766   0.8177620175  19581334192423
2   900   0.8918228764  21185697626083
3   986   0.9209295055  21185697625997
4   1034  0.9113778510  21185697625949
5   1080  0.9011654792  21185697625903
6   1154  0.8975282707  30103357357379
7   1148  0.8849957771  14580922576079
8   790   0.9265178066  11878096933
9   1316  0.9531616349  14580922575911
10  726   0.9509666672  866956873
11  754   0.9409492473  866956873
12  784   0.9363085666  866956873
13  1448  0.9564495245  5995661470529
14  1496  0.9574428891  5995661470481
15  1322  0.9535221550  396016668869
16  1358  0.9465344483  396016668833
17  1688  0.9836927546  8281634108801
18  1722  0.9710521630  8281634108767
19  1812  1.0165154301  8281634108677
20  1830  0.9880814955  8281634108677
21  1844  0.9563187743  8281634108663
22  1680  0.9463064905  968269822189
23  1890  0.9406396232  6200995919731
24  2134  0.9570149690  38986211476747
25  1780  0.9686207607  628177622389
26  2014  0.9341035539  6200995919683
27  1846  0.9534113552  628177622323
28  2088  0.9679949599  3999281381923
29  2116  0.9536970232  3999281381923
30  2400  0.9501210087  38029505632477
31  2478  0.9762139574  38986211476403
32  2524  0.9768240786  38986211476357
33  2560  0.9689295531  38986211476321
34  2286  0.9703645150  2481562496471
35  2320  0.9639271592  2481562496437
36  2616  0.9834171539  17931997861517
37  2396  0.9895774988  1933468592177
38  2444  0.9981020350  1933468592129
39  2472  0.9863866064  1933468592101
40  2538  0.9821956613  2481562496219
41  2760  0.9803005126  10631985435829
42  2380  0.9991966853  327076778191
43  2392  0.9719895984  327076778179
44  2442  0.9873916591  327076778129
45  2470  0.9784290501  327076778101
46  2762  0.9706117929  2481562496219
47  2520  0.9545666043  327076778051
48  2776  0.9415708602  1933468592101
49  3038  0.9415271787  10026387088493
50  3092  0.9531007373  10026387088439
51  2946  0.9460969948  2796148447381
52  2976  0.9382202652  2796148447381
53  3196  0.9187382475  11783179421593
54  3224  0.9279160571  10026387088493
55  3278  0.9396521374  10026387088439
56  3096  0.9237957124  2481562495661
57  3390  0.9461117876  11783179421371
58  3560  0.9395747528  29077945916363
59  3594  0.9376826431  28158788983159
60  3636  0.9343561260  29077945916363
61  3654  0.9164223001  29077945916363
62  3456  0.9287125490  5716399254341
63  3294  0.9469610659  1376589410369
64  3330  0.9464086867  1376589410333
65  3596  0.9378033618  6215409275507
66  3678  0.9740832743  6215409275249
67  3702  0.9617861382  6215409275249
68  3758  0.9762827903  6215409275249
69  3854  1.0242911884  6215409275249
70  3870  1.0052760984  6215409275249
71  3920  1.0147688787  6215409275249
72  3932  0.9927489370  6215409275237
73  3966  0.9891020412  6215409275041
74  4062  1.0366412505  6215409275041
75  4078  1.0180858187  6215409275041
76  4128  1.0276414005  6215409275041
77  4150  1.0142622729  6215409275407
78  4200  1.0238470491  6215409275357
79  4308  1.0809994193  6215409275249
80  4328  1.0659029505  6215409275249
81  4340  1.0444870805  6215409275237
82  4380  1.0459795515  6215409275177
83  4414  1.0426566161  6215409275143
84  4516  1.0944353381  6215409275041
85  4536  1.0796801338  6215409275041
86  4548  1.0586702538  6215409275029
87  4556  1.0347395141  6215409275021
88  4578  1.0221867581  6215409275041
89  4596  1.0066376308  6215409275041
90  4620  0.9959600976  6215409275041
91  4642  0.9838544524  6215409275041
92  5020  0.9684580361  36683716323913
93  5058  0.9781413471  33994032583531
94  5146  1.0006726694  36683716323913
95  5194  1.0063137564  36683716323913
96  5278  1.0371216659  36683716324039
97  5404  1.0977245069  36683716323913
98  5418  1.0792569593  36683716323899
99  5470  1.0876676245  36683716323847
100 5482  1.0680270856  36683716323847
101 5526  1.0708730803  36683716323791
102 5590  1.0876834546  36683716323913
103 5638  1.0933231416  36683716323913
104 5656  1.0781126752  36683716323847
105 5704  1.0837889389  36683716323847
106 5758  1.0936239342  36683716323913
107 5772  1.0758527238  36683716323899
108 5824  1.0843154811  36683716323847
109 5830  1.0612869894  36683716323841

Bonus: some instances CSG > 1 for k <= 1024 and p <= 2*10^12:
210 7700  1.0009864925  185067241757
211 7746  1.0126426509  185067241757
212 7760  1.0003343480  185067241757
213 7790  1.0000214554  185067241757

[Bobby Jacobs]

Originally Posted by mart_r

Data for maximal gaps for p < 3*10¹³ and k <= 109 is now publicly available! Rejoice!
I'm probably taking this up to p = 10¹⁴. Well, unless anyone wants to join in.


Since the primes at the start of a maximal gap almost always* come in clusters, I did a quick check which p_n had the highest number of occurrences for k <= 100, for 3*10¹³ downwards:
* I know that may be a rather daring statement...

#occ   p_n
2      29418557625949  (k = 11, 16)
4      29418557625841  (k = 13, 14, 17, 18)
21     29077945916363  (55 <= k <= 85)
23     1376589410333   (55 <= k <= 87)
30     16025473729     (52 <= k <= 98)
33     3099587         (48 <= k <= 100)
34     18313           (47 <= k <= 95)
39     1621            (24 <= k <= 96)
45     661             (18 <= k <= 100)
52     467             (9 <= k <= 99)
66     283             (6 <= k <= 100)
68     199             (2 <= k <= 96)
73     109             (2 <= k <= 100)
77     7
100    2

2 and 3 always occur as primes preceding maximal gaps. 5 doesn't always occur since for p = 3 (technically p₂ = 3), for some k, p_2+k and p_2+k+1 are twin primes and in that case for p = 5 the gap length is the same as for p = 3. However, whenever 5 doesn't appear as a maximal gap, then 7 definitely does, and with respect to the number of occurrences, 7 is either in the lead by one or ties with 5. No p > 7 appears more often than p = 7 as a prime preceding a maximal gap for k = 1, 2, 3, ..., so p = 7 is a local maximum here.

But let's do this more formally:

Let \(p_n\) be the set of prime numbers and \(o_n(x)\) the set of the number of occurrences of \(p_n\) as primes preceding a maximal gap for all positive integers \(k <= x\).
\(p_n = \{2, 3, 5, 7, 11, ...\}\)
\(o_n(1) = \{1, 1, 0, 1, 0, 0, 0, 0, 1, 0, ...\}\)
\(o_n(1000) = \{1000, 1000, 827, 828, 658, 781, 660, 783, 661, 416, ...\}\)

\(o_n\) and the corresponding \(p_n\) constitutes a local maximum for the above table - in this case for x = 100 - if there does not exist \(m > n\) such that \(o_m(x) > o_n(x)\).

Conjecture: as \(k \to \infty\), the smallest \(p_n\) in the above table with a local maximum of number of occurrences as maximal gap commencers will be fixed. 19 chimes in for a larger range of \(k\), so the list of local maxima \(p_n\) will probably start {2, 7, 19, 109, 199, 9439 (?), ...} for k sufficiently large - this appears to be very tricky, at least numerically...

A follow-up question will be: for fixed x, at what point will the list of local maxima p_n be settled? For example, in the above table for x = 100, could there be a larger p_n preceding a maximal gap for more than half of the values of k (in which case o_n = 45 / p_n = 661 and possibly o_n = 52 / p_n = 467 will be superseded)? Or could there be a gap between consecutive primes so large that all - or at least most - of the p_n for k > 1 also turn out as maximal gaps?

Once creativity strikes... k = 6 is the first k for which p_n = 2, 3, 5, and 7 each start a maximal gap. For k = 12, all of the first five primes appear in the attached list. For k = 19, this makes six primes, and the first 13 (!) primes appear at k = 68 (so p_n+68-p_n becomes continually larger for every p_n <= 41). I bet MattcAnderson would like to see this sequence in the OEIS

I guess I'm biting off more than I can chew...

How many times does 1327 appear in the list? 1327 has some big gaps to the next primes (1361, 1367, 1373, 1381, 1399, 1409, 1423). What about 1321? Since 1321 is near 1327, it should also appear a lot.

[mart_r]

Originally Posted by Bobby Jacobs

How many times does 1327 appear in the list? 1327 has some big gaps to the next primes (1361, 1367, 1373, 1381, 1399, 1409, 1423). What about 1321? Since 1321 is near 1327, it should also appear a lot.

You're right. For small x, 1327 and some of the previous primes should occur quite often as primes preceding maximal gaps. For x >= 8, 1321 occurs more often than 1327, and for x >= 10, 1303 or 1307 occur more often than 1321.

Here's a list for the first 300 primes and the number of occurrences at x = 1000 (i.e. for all k <= 1000) - you clearly see the patterns juxtaposed to the gaps between the consecutive primes:

 p_n  o_n(1000)
   2  1000
   3  1000
   5  827
   7  828
  11  658
  13  781
  17  660
  19  783
  23  661
  29  416
  31  710
  37  408
  41  558
  43  742
  47  658
  53  418
  59  353
  61  687
  67  401
  71  555
  73  741
  79  416
  83  572
  89  406
  97  260
 101  409
 103  664
 107  625
 109  778
 113  669
 127  104
 131  247
 137  254
 139  524
 149  193
 151  433
 157  330
 163  306
 167  497
 173  363
 179  328
 181  653
 191  219
 193  481
 197  568
 199  745
 211  161
 223  84
 227  199
 229  372
 233  476
 239  352
 241  622
 251  216
 257  272
 263  269
 269  285
 271  572
 277  373
 281  541
 283  731
 293  238
 307  76
 311  184
 313  370
 317  470
 331  93
 337  144
 347  90
 349  278
 353  375
 359  304
 367  218
 373  248
 379  258
 383  414
 389  333
 397  239
 401  393
 409  241
 419  144
 421  374
 431  170
 433  409
 439  316
 443  484
 449  368
 457  250
 461  407
 463  667
 467  627
 479  163
 487  159
 491  298
 499  208
 503  345
 509  306
 521  114
 523  353
 541  37
 547  80
 557  60
 563  104
 569  128
 571  296
 577  233
 587  135
 593  179
 599  204
 601  450
 607  308
 613  291
 617  472
 619  667
 631  156
 641  121
 643  317
 647  428
 653  354
 659  320
 661  628
 673  157
 677  328
 683  297
 691  224
 701  142
 709  135
 719  94
 727  106
 733  143
 739  174
 743  303
 751  190
 757  228
 761  373
 769  230
 773  369
 787  88
 797  74
 809  47
 811  158
 821  90
 823  242
 827  332
 829  529
 839  200
 853  65
 857  167
 859  344
 863  431
 877  94
 881  218
 883  445
 887  493
 907  39
 911  115
 919  95
 929  76
 937  90
 941  178
 947  182
 953  197
 967  68
 971  157
 977  175
 983  204
 991  177
 997  206
1009  87
1013  196
1019  208
1021  449
1031  182
1033  404
1039  310
1049  187
1051  416
1061  202
1063  434
1069  335
1087  47
1091  146
1093  314
1097  418
1103  342
1109  325
1117  249
1123  274
1129  285
1151  27
1153  97
1163  76
1171  82
1181  65
1187  96
1193  126
1201  116
1213  58
1217  143
1223  156
1229  185
1231  414
1237  291
1249  112
1259  92
1277  15
1279  70
1283  159
1289  167
1291  352
1297  271
1301  411
1303  600
1307  580
1319  164
1321  424
1327  335
1361  0
1367  7
1373  23
1381  22
1399  2
1409  3
1423  1
1427  9
1429  35
1433  64
1439  54
1447  44
1451  107
1453  227
1459  183
1471  71
1481  58
1483  177
1487  283
1489  439
1493  467
1499  336
1511  123
1523  63
1531  84
1543  49
1549  77
1553  167
1559  172
1567  151
1571  270
1579  190
1583  311
1597  85
1601  190
1607  211
1609  453
1613  486
1619  370
1621  657
1627  406
1637  226
1657  27
1663  59
1667  132
1669  303
1693  11
1697  49
1699  131
1709  89
1721  47
1723  147
1733  83
1741  94
1747  126
1753  158
1759  180
1777  24
1783  57
1787  136
1789  290
1801  99
1811  82
1823  45
1831  54
1847  16
1861  7
1867  18
1871  49
1873  113
1877  174
1879  301
1889  143
1901  75
1907  116
1913  143
1931  23
1933  102
1949  24
1951  93
1973  6
1979  17
1987  21

As one might expect, 1361 has 0 occurrences (the next prime with 0 occurrences for x = 1000 is 2203).
(Note also that 1621 occurs more often than 1303. This is mostly because there are rather many primes between 1400 and 1500 but rather few between 1700 and 1800 as well as between 1800 and 1900.)

The first time p₂₁₈ = 1361 appears as a prime preceding a maximal gap is for k = 1315 because p_217+1315 = p₁₅₃₂ = 12853 and p_218+1315 = p₁₅₃₃ = 12889, which is a gap of 36 between consecutive primes (i.e. more than the 34 between 1327 and 1361) and a gap of 11528 between p₂₁₈ and p₁₅₃₃, while for all n < 218, p_n+1315-p_n < 11528.


If you'd like to play around with a larger set of data, check out the attachment.

[Bobby Jacobs]

Originally Posted by mart_r

Conjecture: as \(k \to \infty\), the smallest \(p_n\) in the above table with a local maximum of number of occurrences as maximal gap commencers will be fixed. 19 chimes in for a larger range of \(k\), so the list of local maxima \(p_n\) will probably start {2, 7, 19, 109, 199, 9439 (?), ...} for k sufficiently large - this appears to be very tricky, at least numerically...

I believe that as \(n\to\infty\), the primes p with the most occurrences will be based upon a lot of small prime gaps immediately before p. Therefore, 5659 should eventually beat 109 because the 5 prime gaps before 5659 are 6, 4, 2, 4, 2, but the 5 prime gaps before 109 are 8, 4, 2, 4, 2.

[mart_r]

Originally Posted by Bobby Jacobs

I believe that as \(n\to\infty\), the primes p with the most occurrences will be based upon a lot of small prime gaps immediately before p. Therefore, 5659 should eventually beat 109 because the 5 prime gaps before 5659 are 6, 4, 2, 4, 2, but the 5 prime gaps before 109 are 8, 4, 2, 4, 2.

p=5659 is not a good candidate for a record number of maximal gaps after p, as you can see in the attached graph. The graph shows p_n vs. o_n(x) at x=500000. Points further to the right have a higher number of occurrences.
5659 is the 746th prime number. o₇₄₆(x)=423464, while for p=9439, we already have o₁₁₇₀(x)=444555.
And, just as an aside, \(\lim_{x\to\infty} x/o_n(x) = 1\) (working out secondary terms will be interesting;).
Whether 9439 would eventually beat 109 remains to be seen...

[mart_r]

9439 beats 283 at around x=740000.
9439 does not appear to beat 199.
113173 may be the subsequent local maximum (beating 24109 for some x < 1.2e6). A lot more o_k and a lot higher bound x would need to be looked at to see whether that remains true.
Note that 113173 is the penultimate number of an almost-decuplet or cousin-nonuplet or whatever you may call it. So Bobby's observation holds true at this point, with my addition that some large gaps directly after such a cluster (or, say, (p-\(\theta\)(p))/\(\sqrt{p}\) is not "too large", YMMV) make for good conditions to produce such "high performer" initial members of these generalized maximal gaps. We may invoke the performance indicator \(\lim_{x\to\infty} \frac{x}{(\log x -1)(x-o_n(x))}\). More sophisticated ideas are welcome.
In principle it might be possible that there exists a larger p that eventually beats 9439, or even 199 or 109 or...??
Intricate problem, delicate computation. Relocate focus? Allocate more resources? Vindicate my existence??

   k    p_k o_k(1e6)
   1      2 1000000
   2      3 1000000
   3      5  913974
   4      7  913975
   5     11  828143
   6     13  901885
   7     17  828145
   8     19  901887
   9     23  828146
  10     29  681628
  11     31  886659
  12     37  680180
  13     41  800714
  14     43  896535
  15     47  827790
  16     53  681558
  17     59  658923
  18     61  883217
  19     67  679222
  20     71  800359
  21     73  896477
  22     79  681232
  23     83  801182
  24     89  678585
  25     97  592056
  26    101  752065
  27    103  889285
  28    107  825738
  29    109  901630
  30    113  828113
  31    127  381766
  32    131  641396
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 864   6701  504185
 865   6703  782017
 866   6709  658984
 867   6719  514820
 868   6733  330626
 869   6737  598769
 870   6761  155922
 871   6763  444468
 872   6779  247825
 873   6781  552518
 874   6791  441278
 875   6793  727156
 876   6803  501444
 877   6823  202920
 878   6827  457719
 879   6829  715507
 880   6833  758735
 881   6841  587145
 882   6857  313259
 883   6863  477205
 884   6869  570412
 885   6871  819870
 886   6883  428021
 887   6899  284549
 888   6907  387801
 889   6911  620493
 890   6917  611911
 891   6947   96715
 892   6949  333305
 893   6959  328100
 894   6961  620277
 895   6967  589051
 896   6971  742396
 897   6977  655567
 898   6983  647537
 899   6991  584163
 900   6997  626478
 901   7001  774720
 902   7013  432865
 903   7019  569419
 904   7027  554772
 905   7039  392107
 906   7043  652035
 907   7057  350636
 908   7069  327327
 909   7079  373455
 910   7103  127442
 911   7109  274851
 912   7121  257577
 913   7127  406696
 914   7129  676893
 915   7151  190092
 916   7159  298240
 917   7177  177273
 918   7187  259746
 919   7193  396678
 920   7207  267741
 921   7211  507210
 922   7213  727592
 923   7219  620679
 924   7229  495545
 925   7237  515224
 926   7243  589911
 927   7247  756526
 928   7253  663797
 929   7283  100673
 930   7297  136380
 931   7307  202696
 932   7309  456225
 933   7321  311782
 934   7331  350593
 935   7333  637481
 936   7349  299132
 937   7351  609703
 938   7369  243694
 939   7393  101571
 940   7411   92086
 941   7417  209683
 942   7433  159480
 943   7451  115373
 944   7457  237882
 945   7459  458353
 946   7477  190972
 947   7481  419437
 948   7487  485089
 949   7489  718144
 950   7499  482623
 951   7507  500708
 952   7517  450680
 953   7523  562536
 954   7529  604751
 955   7537  560954
 956   7541  731018
 957   7547  657245
 958   7549  874039
 959   7559  533420
 960   7561  806029
 961   7573  431111
 962   7577  693945
 963   7583  647800
 964   7589  648250
 965   7591  874560
 966   7603  437384
 967   7607  698495
 968   7621  358362
 969   7639  209942
 970   7643  462514
 971   7649  532784
 972   7669  209623
 973   7673  449842
 974   7681  465751
 975   7687  557140
 976   7691  723945
 977   7699  578417
 978   7703  742997
 979   7717  372437
 980   7723  529536
 981   7727  718094
 982   7741  366036
 983   7753  335167
 984   7757  586816
 985   7759  805664
 986   7789  101290
 987   7793  310708
 988   7817  107529
 989   7823  238258
 990   7829  362089
 991   7841  294758
 992   7853  276436
 993   7867  237917
 994   7873  388145
 995   7877  587133
 996   7879  770904
 997   7883  770281
 998   7901  273698
 999   7907  450006
1000   7919  350255
1001   7927  447230
1002   7933  548613
1003   7937  718802
1004   7949  417802
1005   7951  753233
1006   7963  419300
1007   7993   83831
1008   8009  107456
1009   8011  299059
1010   8017  386665
1011   8039  149072
1012   8053  154876
1013   8059  291640
1014   8069  314601
1015   8081  278252
1016   8087  422275
1017   8089  677719
1018   8093  725639
1019   8101  564015
1020   8111  474186
1021   8117  580411
1022   8123  614559
1023   8147  162735
1024   8161  184935
1025   8167  335236
1026   8171  539215
1027   8179  495718
1028   8191  362521
1029   8209  204301
1030   8219  291170
1031   8221  567386
1032   8231  444250
1033   8233  727774
1034   8237  773312
1035   8243  663068
1036   8263  235989
1037   8269  417567
1038   8273  619215
1039   8287  342851
1040   8291  598860
1041   8293  814501
1042   8297  801132
1043   8311  378146
1044   8317  534439
1045   8329  383350
1046   8353  134075
1047   8363  224221
1048   8369  368955
1049   8377  412013
1050   8387  394147
1051   8389  680917
1052   8419   93879
1053   8423  293000
1054   8429  408254
1055   8431  662652
1056   8443  385953
1057   8447  646304
1058   8461  341841
1059   8467  504484
1060   8501   69186
1061   8513  116949
1062   8521  202233
1063   8527  315343
1064   8537  321914
1065   8539  585546
1066   8543  680144
1067   8563  219319
1068   8573  287906
1069   8581  387302
1070   8597  244204
1071   8599  542087
1072   8609  433592
1073   8623  297926
1074   8627  552710
1075   8629  770476
1076   8641  415269
1077   8647  554279
1078   8663  306473
1079   8669  475503
1080   8677  489320
1081   8681  691717
1082   8689  567613
1083   8693  738977
1084   8699  661015
1085   8707  586511
1086   8713  627340
1087   8719  640503
1088   8731  417157
1089   8737  558055
1090   8741  744102
1091   8747  663209
1092   8753  652517
1093   8761  586277
1094   8779  258670
1095   8783  523790
1096   8803  200263
1097   8807  446712
1098   8819  335205
1099   8821  658715
1100   8831  475262
1101   8837  579208
1102   8839  828177
1103   8849  524644
1104   8861  391986
1105   8863  725742
1106   8867  781045
1107   8887  240885
1108   8893  422612
1109   8923   80478
1110   8929  208909
1111   8933  409046
1112   8941  416152
1113   8951  393585
1114   8963  325890
1115   8969  479337
1116   8971  747736
1117   8999  127243
1118   9001  360626
1119   9007  464505
1120   9011  660889
1121   9013  818627
1122   9029  337805
1123   9041  320627
1124   9043  629664
1125   9049  607921
1126   9059  494063
1127   9067  516754
1128   9091  150623
1129   9103  198792
1130   9109  346340
1131   9127  185103
1132   9133  341617
1133   9137  558076
1134   9151  310334
1135   9157  467711
1136   9161  665337
1137   9173  401783
1138   9181  480776
1139   9187  571163
1140   9199  393381
1141   9203  652871
1142   9209  630918
1143   9221  411875
1144   9227  553895
1145   9239  387908
1146   9241  731940
1147   9257  323508
1148   9277  166449
1149   9281  398003
1150   9283  651804
1151   9293  470813
1152   9311  234415
1153   9319  358320
1154   9323  573906
1155   9337  328114
1156   9341  581934
1157   9343  798460
1158   9349  656105
1159   9371  207995
1160   9377  388085
1161   9391  271152
1162   9397  437220
1163   9403  530335
1164   9413  458656
1165   9419  567130
1166   9421  815835
1167   9431  522055
1168   9433  796818
1169   9437  809480
1170   9439  896807
1171   9461  220160
1172   9463  526660
1173   9467  691318
1174   9473  640024
1175   9479  642888
1176   9491  417162
1177   9497  557734
1178   9511  344895
1179   9521  380990
1180   9533  331294
1181   9539  485629
1182   9547  508534
1183   9551  691160
1184   9587   63385
1185   9601   95979
1186   9613  130162
1187   9619  247563
1188   9623  418077
1189   9629  464520
1190   9631  708552
1191   9643  389335
1192   9649  529232
1193   9661  374089
1194   9677  261811
1195   9679  562329
1196   9689  453557
1197   9697  488792
1198   9719  176513
1199   9721  459824
1200   9733  333283
1201   9739  482070
1202   9743  683872
1203   9749  633336
1204   9767  263156
1205   9769  602146
1206   9781  381785
1207   9787  531735
1208   9791  718107
1209   9803  419918
1210   9811  495878
1211   9817  583689
1212   9829  399246
1213   9833  659599
1214   9839  635254
1215   9851  413756
1216   9857  555795
1217   9859  826057
1218   9871  429932
1219   9883  362286
1220   9887  633945
1221   9901  341356
1222   9907  506371
1223   9923  290652
1224   9929  457519
1225   9931  736097
1226   9941  501827
1227   9949  519633
1228   9967  243664
1229   9973  418661
1230  10007   63093
1231  10009  226219
1232  10037   64385
1233  10039  212523
1234  10061   96217
1235  10067  212622
(...)
2684  24109  889952
:727 113173  889409

For these k, the first n primes are preceding generalized maximal gaps p_n+k-p_n:

  n  k
  2  1
  3  2
  4  6
  5  12
  6  19
  7  97
  8  70
  9  120
 10  88
 11  119
 12  237
 13  68
 14  681
 15  412
 16  1591
 17  2907
 18  1510
 19  2734
 20  2131
 21  1588
 22  3834
 23  6041
 24  2897
 25  11562
 26  21004
 27  11560
 28  44194
 29  21001
 30  11557
 31  25174
 32  32114
 33  131271
 34  36918
 35  44636
 36  115242
 37  211442
 38  477957
 39  64935
 40  204412
 41  710665
 42  175930
 43  438049
 44  409641
 45  725804
 46  176350
 47  560510
 48  2570641
 49  2841381
 50  4094784
 51  1063896
 52  4355669
 53  1807346
 54  2070798
 55  2349691
 56  6380527
 57  6563887
 58  6276812
 59  14215737
 60  8543349
 61  2899899
 62  7714640
 63  19264207
 64  15644556
 65  13668980
 66  10701209
 67  24451150
 68  13668996
 69  38417236
 70  33907310
 71  25958214
 72  37376935
 73  72210305
 74  51624533
 75  155807588
 76  121101282
 77  72019160
 78  199395703
 79  34335444
 80  80104183
 81  575130837
 82  273221126
 83  362546538
 84  478749161
 85  209832527
 86  92967699
 87  251653222
 90  833367050
 91  566487675
 92  212341969
 93  838711510
 94  394795699
 97  457331290
 99  864115614
107  834990586

Search limit: k=9e8

And now for the cherry on top of it:
For 25698372294281 <= p <= 25698372297167 there are 144 values of k with 302 <= k <= 445 for which a new CSG maximum is > 1, with the largest instance at p = 25698372297029, k = 316, CSG = 1.09729237...

Ah, the fun we have

[mart_r]

Originally Posted by mart_r

Relocate focus?

That's what. You know, even though I don't get many replies, it helps that I share my ideas here as it puts more pressure on me to think things through more thoroughly (try saying that five times fast:), beneath all my rampant numerology.

Originally Posted by Bobby Jacobs

I believe that as \(n\to\infty\), the primes p with the most occurrences will be based upon a lot of small prime gaps immediately before p. Therefore, 5659 should eventually beat 109 because the 5 prime gaps before 5659 are 6, 4, 2, 4, 2, but the 5 prime gaps before 109 are 8, 4, 2, 4, 2.

That seems to be right after all - I stand corrected. Those "high performer" primes preceding maximal gaps depend primarily on the small gaps right before them. I can see it now - it might be well out of reach for an actual computation, but on an asymptotic scale, 5659, being the last member of a prime-septuplet, does have a good chance to beat 109 sometime.

[Bobby Jacobs]

What is the pattern with the sequence of primes with record low numbers of occurrences? It seems like the sequence is 2, 5, 11, 29, 37, 59, 97, 127, 223, 307, 541, 907, 1151, 1361, ... This is similar to the primes at the end of maximal prime gaps, but not exactly. I wonder what the pattern is.

[mart_r]

Me too
At first sight, 37 should occur more often than 29 because the two gaps preceding 37 are {2, 6} instead of {4, 6} for 29. If however we take three gaps before the prime into account, it's {6, 2, 6} vs. {2, 4, 6}. The {2, 4, 6}-pattern having more open residues mod 5 also plays a role, favoring 37 as a local record minimum in number of occurrences. Now, at what margin remains 37 below 29?