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-   -   Gaps between non-consecutive primes (https://www.mersenneforum.org/showthread.php?t=27301)

robert44444uk 2022-01-24 09:11

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.[/code] 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 2022-03-10 21:16

Herr Ober, Zahlen bitte!
 
1 Attachment(s)
Data for maximal gaps for p < 3*10[SUP]13[/SUP] and k <= 109 is now publicly available! Rejoice!
I'm probably taking this up to p = 10[SUP]14[/SUP]. 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[SUB]n[/SUB] had the highest number of occurrences for k <= 100, for 3*10[SUP]13[/SUP] downwards:
[SIZE=1]* I know that may be a rather daring statement...[/SIZE]

[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[/CODE]2 and 3 always occur as primes preceding maximal gaps. 5 doesn't always occur since for p = 3 (technically p[SUB]2[/SUB] = 3), for some k, p[SUB]2+k[/SUB] and p[SUB]2+k+1[/SUB] 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 [I]very[/I] tricky, at least numerically...

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k (in which case o[SUB]n[/SUB] = 45 / p[SUB]n[/SUB] = 661 and possibly o[SUB]n[/SUB] = 52 / p[SUB]n[/SUB] = 467 will be superseded)?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

Once creativity strikes... k = 6 is the first k for which p[SUB]n[/SUB] = 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[SUB]n+68[/SUB]-p[SUB]n[/SUB] becomes continually larger for every p[SUB]n[/SUB] <= 41). I bet MattcAnderson would like to see this sequence in the OEIS :wink:

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

mart_r 2022-03-16 22:06

[QUOTE=Bobby Jacobs;592658]For each k, what are the first few gaps with record CSG ratio? This is very interesting.[/QUOTE]
These are the current record CSG for each k @ p <= 3.9*10[SUP]13[/SUP]:
[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[/CODE]

Bobby Jacobs 2022-03-20 20:33

[QUOTE=mart_r;601471]Data for maximal gaps for p < 3*10[SUP]13[/SUP] and k <= 109 is now publicly available! Rejoice!
I'm probably taking this up to p = 10[SUP]14[/SUP]. 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[SUB]n[/SUB] had the highest number of occurrences for k <= 100, for 3*10[SUP]13[/SUP] downwards:
[SIZE=1]* I know that may be a rather daring statement...[/SIZE]

[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[/CODE]2 and 3 always occur as primes preceding maximal gaps. 5 doesn't always occur since for p = 3 (technically p[SUB]2[/SUB] = 3), for some k, p[SUB]2+k[/SUB] and p[SUB]2+k+1[/SUB] 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 [I]very[/I] tricky, at least numerically...

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

Once creativity strikes... k = 6 is the first k for which p[SUB]n[/SUB] = 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[SUB]n+68[/SUB]-p[SUB]n[/SUB] becomes continually larger for every p[SUB]n[/SUB] <= 41). I bet MattcAnderson would like to see this sequence in the OEIS :wink:

I guess I'm biting off more than I can chew... :smile:[/QUOTE]

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 2022-03-21 22:08

1 Attachment(s)
[QUOTE=Bobby Jacobs;602186]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.[/QUOTE]

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
[/CODE]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 p[SUB]218[/SUB] = 1361 appears as a prime preceding a maximal gap is for k = 1315 because p[SUB]217+1315[/SUB] = p[SUB]1532[/SUB] = 12853 and p[SUB]218+1315[/SUB] = p[SUB]1533[/SUB] = 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 p[SUB]218[/SUB] and p[SUB]1533[/SUB], while for all n < 218, p[SUB]n+1315[/SUB]-p[SUB]n[/SUB] < 11528.


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

Bobby Jacobs 2022-03-30 16:55

[QUOTE=mart_r;601471]
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 [I]very[/I] tricky, at least numerically...
[/QUOTE]

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 2022-04-13 20:26

1 Attachment(s)
[QUOTE=Bobby Jacobs;602880]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.[/QUOTE]

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[SUB]n[/SUB] vs. o[SUB]n[/SUB](x) at x=500000. Points further to the right have a higher number of occurrences.
5659 is the 746th prime number. o[SUB]746[/SUB](x)=423464, while for p=9439, we already have o[SUB]1170[/SUB](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 2022-04-22 17:11

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 o[SUB]k[/SUB] 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? [COLOR="LemonChiffon"]Vindicate my existence??[/COLOR]

[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
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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
[/CODE]

For these k, the first n primes are preceding generalized maximal gaps p[SUB]n+k[/SUB]-p[SUB]n[/SUB]:
[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
[/CODE]

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 :smile:

mart_r 2022-04-23 15:59

[QUOTE=mart_r;604557]Relocate focus?[/QUOTE]

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=Bobby Jacobs;602880]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.[/QUOTE]

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 2022-04-25 19:08

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 2022-04-26 09:18

Me too :smile:
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?


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