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 2022-01-24, 09:11 #34 robert44444uk     Jun 2003 Suva, Fiji 7F816 Posts 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 Code: 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. Last fiddled with by Dr Sardonicus on 2022-01-24 at 14:41 Reason: Add code tags for jillion-digit number
2022-03-10, 21:16   #35
mart_r

Dec 2008
you know...around...

24×53 Posts
Herr Ober, Zahlen bitte!

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 pn had the highest number of occurrences for k <= 100, for 3*1013 downwards:
* I know that may be a rather daring statement...

Code:
#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, p2+k and p2+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 pn be settled? For example, in the above table for x = 100, could there be a larger pn preceding a maximal gap for more than half of the values of k (in which case on = 45 / pn = 661 and possibly on = 52 / pn = 467 will be superseded)? Or could there be a gap between consecutive primes so large that all - or at least most - of the pn for k > 1 also turn out as maximal gaps?

Once creativity strikes... k = 6 is the first k for which pn = 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 pn+68-pn becomes continually larger for every pn <= 41). I bet MattcAnderson would like to see this sequence in the OEIS

I guess I'm biting off more than I can chew...
Attached Files
 GNCP_maxgaps_3e13.zip (127.7 KB, 65 views)

2022-03-16, 22:06   #36
mart_r

Dec 2008
you know...around...

24×53 Posts

Quote:
 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:
Code:
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

2022-03-20, 20:33   #37
Bobby Jacobs

May 2018

23·5·7 Posts

Quote:
 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 pn had the highest number of occurrences for k <= 100, for 3*1013 downwards: * I know that may be a rather daring statement... Code: #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, p2+k and p2+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 pn be settled? For example, in the above table for x = 100, could there be a larger pn preceding a maximal gap for more than half of the values of k (in which case on = 45 / pn = 661 and possibly on = 52 / pn = 467 will be superseded)? Or could there be a gap between consecutive primes so large that all - or at least most - of the pn for k > 1 also turn out as maximal gaps? Once creativity strikes... k = 6 is the first k for which pn = 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 pn+68-pn becomes continually larger for every pn <= 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.

2022-03-21, 22:08   #38
mart_r

Dec 2008
you know...around...

15208 Posts

Quote:
 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:
Code:
 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 p217+1315 = p1532 = 12853 and p218+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, pn+1315-pn < 11528.

If you'd like to play around with a larger set of data, check out the attachment.
Attached Files
 Bobby.zip (185.2 KB, 55 views)

2022-03-30, 16:55   #39
Bobby Jacobs

May 2018

11816 Posts

Quote:
 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.

2022-04-13, 20:26   #40
mart_r

Dec 2008
you know...around...

24×53 Posts

Quote:
 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 pn vs. on(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...
Attached Thumbnails

 2022-04-22, 17:11 #41 mart_r     Dec 2008 you know...around... 11010100002 Posts What do you get if you multiply six by nine? 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 ok 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?? Code:  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 33 137 629027 34 139 864356 35 149 532451 36 151 807001 37 157 668832 38 163 655002 39 167 793753 40 173 676703 41 179 657575 42 181 882331 43 191 535553 44 193 808696 45 197 814221 46 199 899274 47 211 440323 48 223 366557 49 227 639491 50 229 823828 51 233 811478 52 239 678221 53 241 884641 54 251 536345 55 257 619570 56 263 637297 57 269 645947 58 271 875046 59 277 678153 60 281 799895 61 283 896333 62 293 538088 63 307 340314 64 311 607258 65 313 814055 66 317 807355 67 331 378875 68 337 535249 69 347 461147 70 349 761814 71 353 795136 72 359 673490 73 367 590485 74 373 629575 75 379 641650 76 383 785544 77 389 674505 78 397 590903 79 401 751480 80 409 594445 81 419 489062 82 421 784242 83 431 519907 84 433 800681 85 439 667121 86 443 795923 87 449 677548 88 457 591710 89 461 751923 90 463 889212 91 467 825736 92 479 440288 93 487 511336 94 491 705113 95 499 578974 96 503 746090 97 509 664344 98 521 421689 99 523 763580 100 541 271180 101 547 450118 102 557 429483 103 563 551413 104 569 601162 105 571 843445 106 577 669076 107 587 517671 108 593 609227 109 599 633704 110 601 866713 111 607 676096 112 613 657193 113 617 794421 114 619 894991 115 631 440101 116 641 449185 117 643 747316 118 647 793640 119 653 673934 120 659 656689 121 661 881502 122 673 438065 123 677 699049 124 683 649989 125 691 579297 126 701 484408 127 709 518248 128 719 461765 129 727 503357 130 733 585160 131 739 619697 132 743 774027 133 751 594239 134 757 631679 135 761 779383 136 769 597441 137 773 756587 138 787 375604 139 797 405533 140 809 343311 141 811 677349 142 821 485015 143 823 773652 144 827 798438 145 829 893732 146 839 537224 147 853 340043 148 857 606855 149 859 813837 150 863 807195 151 877 378877 152 881 638598 153 883 841158 154 887 813505 155 907 244026 156 911 503828 157 919 497066 158 929 443645 159 937 494496 160 941 691894 161 947 644162 162 953 644277 163 967 367061 164 971 621546 165 977 619750 166 983 634068 167 991 581393 168 997 624884 169 1009 412527 170 1013 674085 171 1019 641875 172 1021 866845 173 1031 533531 174 1033 807228 175 1039 668959 176 1049 518833 177 1051 798028 178 1061 526147 179 1063 802249 180 1069 667699 181 1087 271076 182 1091 552183 183 1093 776155 184 1097 796662 185 1103 674267 186 1109 656548 187 1117 588379 188 1123 628691 189 1129 641397 190 1151 206487 191 1153 504779 192 1163 433703 193 1171 476260 194 1181 442801 195 1187 560904 196 1193 605981 197 1201 562730 198 1213 393293 199 1217 655929 200 1223 631357 201 1229 641244 202 1231 871744 203 1237 676947 204 1249 424889 205 1259 442089 206 1277 230001 207 1279 537647 208 1283 688222 209 1289 635527 210 1291 860410 211 1297 673956 212 1301 797723 213 1303 895422 214 1307 827464 215 1319 440624 216 1321 774222 217 1327 659014 218 1361 79813 219 1367 209066 220 1373 340459 221 1381 389501 222 1399 198160 223 1409 279261 224 1423 223083 225 1427 455166 226 1429 679754 227 1433 730388 228 1439 642468 229 1447 572724 230 1451 738132 231 1453 879575 232 1459 676456 233 1471 424651 234 1481 441874 235 1483 737786 236 1487 790351 237 1489 889188 238 1493 826028 239 1499 681223 240 1511 425957 241 1523 358233 242 1531 460726 243 1543 356252 244 1549 512475 245 1553 700593 246 1559 644827 247 1567 578889 248 1571 743381 249 1579 591731 250 1583 752116 251 1597 375099 252 1601 632095 253 1607 625731 254 1609 862911 255 1613 817332 256 1619 679959 257 1621 886123 258 1627 680141 259 1637 521397 260 1657 208206 261 1663 390702 262 1667 604307 263 1669 802036 264 1693 170780 265 1697 423178 266 1699 663931 267 1709 484307 268 1721 374835 269 1723 704772 270 1733 497014 271 1741 517825 272 1747 595137 273 1753 623020 274 1759 639435 275 1777 265861 276 1783 442245 277 1787 669006 278 1789 830933 279 1801 430556 280 1811 444024 281 1823 361014 282 1831 445895 283 1847 273720 284 1861 233467 285 1867 400434 286 1871 610121 287 1873 793294 288 1877 789955 289 1879 885866 290 1889 535359 291 1901 396111 292 1907 544980 293 1913 602869 294 1931 258459 295 1933 599789 296 1949 296307 297 1951 619466 298 1973 188985 299 1979 364879 300 1987 421153 301 1993 519458 302 1997 705400 303 1999 849964 304 2003 811377 305 2011 600373 306 2017 633940 307 2027 506154 308 2029 791754 309 2039 524056 310 2053 335861 311 2063 383029 312 2069 517738 313 2081 376196 314 2083 709380 315 2087 768481 316 2089 882615 317 2099 534823 318 2111 396210 319 2113 730259 320 2129 325708 321 2131 650994 322 2137 619950 323 2141 773393 324 2143 885306 325 2153 536756 326 2161 539390 327 2179 248779 328 2203 103983 329 2207 301353 330 2213 407594 331 2221 435477 332 2237 261184 333 2239 564008 334 2243 688647 335 2251 556641 336 2267 304289 337 2269 627988 338 2273 733979 339 2281 578974 340 2287 623886 341 2293 637550 342 2297 783865 343 2309 435095 344 2311 769225 345 2333 208722 346 2339 389671 347 2341 664115 348 2347 616289 349 2351 767295 350 2357 667379 351 2371 369003 352 2377 528523 353 2381 714310 354 2383 868899 355 2389 675807 356 2393 798560 357 2399 677930 358 2411 425375 359 2417 563845 360 2423 612280 361 2437 360831 362 2441 614207 363 2447 616654 364 2459 408719 365 2467 494829 366 2473 582726 367 2477 744373 368 2503 149821 369 2521 122824 370 2531 198653 371 2539 286461 372 2543 482322 373 2549 516578 374 2551 763367 375 2557 633623 376 2579 202834 377 2591 238824 378 2593 527746 379 2609 275415 380 2617 373286 381 2621 601298 382 2633 384112 383 2647 293422 384 2657 345619 385 2659 651972 386 2663 729281 387 2671 576743 388 2677 619995 389 2683 635299 390 2687 781478 391 2689 889819 392 2693 825487 393 2699 680961 394 2707 592895 395 2711 752225 396 2713 889327 397 2719 679804 398 2729 521338 399 2731 800345 400 2741 526548 401 2749 533833 402 2753 727666 403 2767 369050 404 2777 402203 405 2789 341725 406 2791 673881 407 2797 619118 408 2801 768669 409 2803 884785 410 2819 348263 411 2833 267563 412 2837 536535 413 2843 576410 414 2851 544850 415 2857 604937 416 2861 762727 417 2879 276501 418 2887 403023 419 2897 398584 420 2903 524835 421 2909 587106 422 2917 555130 423 2927 470360 424 2939 368898 425 2953 291244 426 2957 539196 427 2963 578023 428 2969 609911 429 2971 853615 430 2999 135663 431 3001 377224 432 3011 375081 433 3019 435537 434 3023 646243 435 3037 344138 436 3041 600602 437 3049 540405 438 3061 384483 439 3067 534120 440 3079 380813 441 3083 646399 442 3089 628391 443 3109 229125 444 3119 298909 445 3121 609687 446 3137 296193 447 3163 96427 448 3167 273277 449 3169 522994 450 3181 337380 451 3187 485383 452 3191 677429 453 3203 404609 454 3209 544594 455 3217 538341 456 3221 712941 457 3229 581473 458 3251 191377 459 3253 493381 460 3257 662046 461 3259 827402 462 3271 428970 463 3299 110859 464 3301 328491 465 3307 439934 466 3313 523151 467 3319 577109 468 3323 744830 469 3329 657443 470 3331 871040 471 3343 436091 472 3347 697601 473 3359 419318 474 3361 749465 475 3371 511805 476 3373 794333 477 3389 339942 478 3391 663593 479 3407 317447 480 3413 485373 481 3433 196669 482 3449 177667 483 3457 283750 484 3461 496750 485 3463 710739 486 3467 748381 487 3469 864755 488 3491 216373 489 3499 325129 490 3511 295169 491 3517 454675 492 3527 426576 493 3529 706953 494 3533 768968 495 3539 662261 496 3541 874313 497 3547 677090 498 3557 520352 499 3559 799201 500 3571 429423 501 3581 444069 502 3583 742622 503 3593 511697 504 3607 331423 505 3613 490540 506 3617 698211 507 3623 643396 508 3631 578801 509 3637 623427 510 3643 638642 511 3659 328517 512 3671 316903 513 3673 628232 514 3677 738302 515 3691 365020 516 3697 523418 517 3701 714872 518 3709 576977 519 3719 482730 520 3727 516720 521 3733 594282 522 3739 624150 523 3761 203934 524 3767 384383 525 3769 656480 526 3779 481453 527 3793 319156 528 3797 583260 529 3803 599607 530 3821 256905 531 3823 594116 532 3833 455338 533 3847 314010 534 3851 573882 535 3853 792200 536 3863 514446 537 3877 333267 538 3881 598369 539 3889 539793 540 3907 246959 541 3911 510676 542 3917 568363 543 3919 813421 544 3923 802319 545 3929 675323 546 3931 883218 547 3943 438508 548 3947 699625 549 3967 230279 550 3989 120105 551 4001 167646 552 4003 414702 553 4007 561937 554 4013 564455 555 4019 595016 556 4021 835490 557 4027 663485 558 4049 209084 559 4051 506397 560 4057 548296 561 4073 305295 562 4079 473101 563 4091 359147 564 4093 684769 565 4099 623089 566 4111 409079 567 4127 276614 568 4129 583433 569 4133 719779 570 4139 650097 571 4153 364128 572 4157 622098 573 4159 832491 574 4177 277476 575 4201 111320 576 4211 199920 577 4217 341369 578 4219 586705 579 4229 439823 580 4231 719258 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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 pn+k-pn: Code:  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 Last fiddled with by mart_r on 2022-04-22 at 17:23 Reason: sopyt gnixif
2022-04-23, 15:59   #42
mart_r

Dec 2008
you know...around...

24·53 Posts

Quote:
 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.

Quote:
 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.

 2022-04-25, 19:08 #43 Bobby Jacobs     May 2018 23·5·7 Posts 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.
 2022-04-26, 09:18 #44 mart_r     Dec 2008 you know...around... 24·53 Posts 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? Last fiddled with by mart_r on 2022-04-26 at 09:27 Reason: little less verbiage

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