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Bobby Jacobs 2022-04-29 23:22

Forbidden prime gap combinations
 
Let an n-prime gap be the gap between a prime p and the prime n primes after p. Then, 2 and 3 are always the start of a maximal n-gap for all n. 5 is the start of a maximal n-gap if and only if the (n+2)nd prime and the (n+3)rd prime are not twin primes. 7 is the start of a maximal n-gap if and only if the (n+3)rd prime and the (n+4)th prime are not twin primes. 11 is the start of a maximal n-gap if and only if the (n+4)th and (n+5)th primes have a gap greater than 4. 13 is the start of a maximal n-gap if and only if the (n+5)th and (n+6)th primes are not twin primes, and the last 2 gaps before the (n+6)th prime are not (2, 4). Basically, every prime has a set of "forbidden prime gap combinations" such that the mth prime is the start of a maximal n-gap if and only if the last gaps before the (m+n)th prime are not one of the forbidden gap combinations. Here are the forbidden gap combinations of the first few primes.

[CODE]
2
[]
3
[]
5
[[2]]
7
[[2]]
11
[[2], [4]]
13
[[2], [2, 4]]
17
[[2], [4]]
19
[[2], [2, 4]]
23
[[2], [4]]
29
[[2], [4], [6]]
31
[[2], [2, 4], [2, 6], [2, 6, 4], [2, 4, 6]]
37
[[2], [4], [6], [2, 4, 8], [2, 4, 2, 10]]
41
[[2], [4], [2, 6], [4, 6], [2, 4, 6, 6], [4, 2, 4, 8], [2, 4, 2, 10]]
43
[[2], [2, 4], [2, 6, 4], [4, 2, 6], [2, 4, 6], [2, 4, 6, 2, 6], [2, 4, 2, 4, 8]]
47
[[2], [4], [4, 2, 4, 6], [4, 2, 4, 6, 2, 6], [4, 2, 4, 2, 4, 8]]
[/CODE]

Notice that 29 just has the forbidden gaps 2, 4, 6, but 37 has the extra combinations (2, 4, 8) and (2, 4, 2, 10). That is why 29 is more common than 37.

mart_r 2022-05-01 14:56

[QUOTE=Bobby Jacobs;604986]Here are the forbidden gap combinations of the first few primes.
[/QUOTE]

Very good! That's the sort of analysis I was looking for.
Do you have a program for these gap combinations?

Bobby Jacobs 2022-05-11 16:28

Yes. I have a program, but it is slow for primes above 47. We basically want admissible k-tuples where the total of the gaps is less than or equal to the total of the k gaps before p. Let p[SUB]m[/SUB] be the mth prime. Suppose the k gaps before the (m+n)th prime are one of these forbidden k-tuples. If p[SUB]m+n[/SUB]-p[SUB]m+n-k[/SUB]<=p[SUB]m[/SUB]-p[SUB]m-k[/SUB], then p[SUB]m+n-k[/SUB]-p[SUB]m-k[/SUB]>=p[SUB]m+n[/SUB]-p[SUB]m[/SUB]. Then, the (m-k)th prime will have at least as big of an n-gap as the mth prime. Therefore, the forbidden gaps are minimal admissible k-tuples >= the k gaps before p[SUB]m[/SUB].

mart_r 2022-05-17 17:02

CSG[SUB]max[/SUB] for p<=10[SUP]14[/SUP]:
[CODE] k gap CSG_max p
1 766 0.81776202 19581334192423
2 900 0.89182288 21185697626083
3 986 0.92092951 21185697625997
4 1134 0.93874248 66592576389587
5 1170 0.91718026 66592576389551
6 1154 0.89752827 30103357357379
7 1148 0.88499578 14580922576079
8 790 0.92651781 11878096933
9 1316 0.95316163 14580922575911
10 726 0.95096666 866956873
11 754 0.94094924 866956873
12 784 0.93630856 866956873
13 1448 0.95644952 5995661470529
14 1496 0.95744289 5995661470481
15 1322 0.95352216 396016668869
16 1358 0.94653445 396016668833
17 1688 0.98369275 8281634108801
18 1722 0.97105216 8281634108767
19 1812 1.01651543 8281634108677
20 1830 0.98808150 8281634108677
21 2134 1.02168813 78736011999913
22 2148 0.99072269 78736011999913
23 2166 0.96394446 78736011999913
24 2310 1.04764008 78736011999913
25 2322 1.01591301 78736011999901
26 2338 0.98829568 78736011999913
27 2376 0.98009540 78736011999847
28 2432 0.98752862 78736011999791
29 2454 0.96623635 78736011999769
30 2494 0.96053115 78736011999913
31 2478 0.97621396 38986211476403
32 2524 0.97682408 38986211476357
33 2560 0.96892955 38986211476321
34 2286 0.97036452 2481562496471
35 2320 0.96392716 2481562496437
36 2616 0.98341715 17931997861517
37 2396 0.98957750 1933468592177
38 2444 0.99810203 1933468592129
39 2472 0.98638661 1933468592101
40 2538 0.98219566 2481562496219
41 2760 0.98030051 10631985435829
42 2380 0.99919669 327076778191
43 2392 0.97198960 327076778179
44 2442 0.98739166 327076778129
45 2470 0.97842905 327076778101
46 2762 0.97061179 2481562496219
47 2520 0.95456660 327076778051
48 2776 0.94157086 1933468592101
49 3038 0.94152718 10026387088493
50 3092 0.95310074 10026387088439
51 2946 0.94609699 2796148447381
52 2976 0.93822027 2796148447381
53 3450 0.93208471 60681682061173
54 3224 0.92791606 10026387088493
55 3278 0.93965214 10026387088439
56 3096 0.92379571 2481562495661
57 3390 0.94611179 11783179421371
58 3560 0.93957475 29077945916363
59 3808 0.96141677 90210824580841
60 3764 0.95339422 55956455554739
61 3798 0.94719704 55956455554739
62 3852 0.95602954 55956455554651
63 3942 0.99181087 55956455554561
64 3976 0.98566033 55956455554561
65 4004 1.00012038 45921691543349
66 4020 0.98072956 45921691543333
67 4086 0.99893031 45921691543267
68 4140 1.00814094 45921691543213
69 3854 1.02429119 6215409275249
70 4292 1.05955757 45921691543061
71 4310 1.04178765 45921691543043
72 4332 1.02721666 45921691543061
73 4386 1.03648387 45921691543061
74 4062 1.03664125 6215409275041
75 4078 1.01808582 6215409275041
76 4128 1.02764140 6215409275041
77 4150 1.01426227 6215409275407
78 4200 1.02384705 6215409275357
79 4308 1.08099942 6215409275249
80 4328 1.06590295 6215409275249
81 4340 1.04448708 6215409275237
82 4380 1.04597955 6215409275177
83 4414 1.04265662 6215409275143
84 4516 1.09443534 6215409275041
85 4536 1.07968013 6215409275041
86 4548 1.05867025 6215409275029
87 4556 1.03473951 6215409275021
88 4578 1.02218676 6215409275041
89 4596 1.00663763 6215409275041
90 4620 0.99596010 6215409275041
91 4642 0.98385445 6215409275041
92 5020 0.96845804 36683716323913
93 5058 0.97814135 33994032583531
94 5146 1.00067267 36683716323913
95 5194 1.00631376 36683716323913
96 5278 1.03712167 36683716324039
97 5404 1.09772451 36683716323913
98 5418 1.07925696 36683716323899
99 5470 1.08766762 36683716323847
100 5482 1.06802709 36683716323847
101 5526 1.07087308 36683716323791
102 5590 1.08768345 36683716323913
103 5638 1.09332314 36683716323913
104 5656 1.07811268 36683716323847
105 5704 1.08378894 36683716323847
106 5758 1.09362393 36683716323913
107 5772 1.07585272 36683716323899
108 5824 1.08431548 36683716323847
109 5830 1.06128699 36683716323841
[/CODE]

And just above 10[SUP]14[/SUP], these 22 new records showed up:
[CODE] 10 1528 0.96314466 102591551174059
11 1560 0.94298881 102591551174027
50 3450 0.97333053 102267713449991
51 3480 0.96260938 102267713449991
52 3562 0.99122668 102267713449879
53 3592 0.98063297 102267713449879
54 3634 0.97918812 102267713449807
55 3684 0.98379105 102267713449757
56 3714 0.97357591 102267713449757
57 3768 0.98125523 102267713449673
58 3798 0.97126377 102267713449673
59 3834 0.96582204 102267713449607
60 3874 0.96340363 102267713449567
61 3904 0.95381038 102267713449567
62 3958 0.96169379 102267713449483
66 4186 1.00199403 102267713449117
68 4324 1.03945196 102267713449117
69 4354 1.03013486 102267713449117
76 4658 1.03478754 101562452774609
77 4694 1.03029216 101562452774609
92 5304 1.01634058 102267713449117
93 5328 1.00471893 102267713449093
[/CODE]

@ Bobby: I'm working on a program to look for the forbidden gap combinations. If it works, it should be fast enough for primes up to at least 97 (well at least I hope so).

mart_r 2022-05-19 17:28

It appears my VBA code for "forbidden gap combinations" (for getting a heuristic grip on the generalized maximal gap candidates) works as it should:
[CODE] 5: [ 2]
7: [ 2]
11: [ 2], [ 4]
13: [ 2], [ 2, 4]
17: [ 2], [ 4]
19: [ 2], [ 2, 4]
23: [ 2], [ 4]
29: [ 2], [ 4], [ 6]
31: [ 2], [ 2, 4], [ 2, 6], [ 2, 6, 4], [ 2, 4, 6]
37: [ 2], [ 4], [ 6], [ 2, 4, 8], [ 2, 4, 2, 10]
41: [ 2], [ 4], [ 2, 6], [ 4, 6], [ 2, 4, 6, 6], [ 4, 2, 4, 8], [ 2, 4, 2, 10]
43: [ 2], [ 2, 4], [ 2, 6, 4], [ 4, 2, 6], [ 2, 4, 6], [ 2, 4, 6, 2, 6], [ 2, 4, 2, 4, 8]
47: [ 2], [ 4], [ 4, 2, 4, 6], [ 4, 2, 4, 6, 2, 6], [ 4, 2, 4, 2, 4, 8]
53: [ 2], [ 4], [ 6], [ 2, 4, 2, 4, 8], [ 4, 2, 4, 2, 10], [ 2, 4, 2, 4, 6, 2, 10]
59: [ 2], [ 4], [ 6], [ 4, 8], [ 2, 10], [ 2, 4, 2, 4, 6, 8], [ 2, 4, 2, 4, 6, 2, 6, 10]
61: [ 2], [ 2, 4], [ 2, 6], [ 2, 6, 4], [ 2, 4, 6], [ 2, 6, 6], [ 2, 4, 8], [ 2, 6, 6, 4], [ 2, 6, 4, 6], [ 2, 4, 6, 6], [ 2, 4, 2, 10], [ 2, 4, 2, 4, 8, 6, 4], [ 2, 4, 2, 4, 6, 8, 4], [ 4, 2, 4, 2, 4, 8, 6], [ 2, 4, 2, 4, 6, 2, 10], [ 2, 4, 2, 4, 6, 2, 6, 6, 4, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 6, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 10]
67: [ 2], [ 4], [ 6], [ 2, 4, 8], [ 2, 6, 4, 8], [ 2, 4, 6, 8], [ 2, 4, 2, 10], [ 2, 4, 2, 12], [ 2, 6, 4, 2, 10], [ 2, 4, 6, 2, 10], [ 2, 4, 2, 4, 6, 2, 6, 10], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 12]
71: [ 2], [ 4], [ 2, 6], [ 4, 6], [ 2, 4, 6, 6], [ 4, 2, 4, 8], [ 2, 4, 2, 10], [ 4, 6, 2, 6, 6], [ 2, 6, 4, 6, 6], [ 4, 2, 4, 8, 6], [ 4, 6, 2, 4, 8], [ 4, 2, 6, 4, 8], [ 4, 2, 4, 6, 8], [ 2, 6, 4, 2, 10], [ 2, 4, 6, 2, 10], [ 4, 2, 4, 2, 12], [ 2, 4, 2, 4, 6, 2, 6, 6, 4, 6, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 6, 6, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 10, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 12]
73: [ 2], [ 2, 4], [ 2, 6, 4], [ 4, 2, 6], [ 2, 4, 6], [ 2, 4, 6, 2, 6], [ 2, 4, 2, 4, 8], [ 2, 6, 4, 6, 2, 6], [ 2, 4, 6, 6, 2, 6], [ 2, 4, 2, 10, 2, 6], [ 2, 4, 6, 2, 6, 6], [ 2, 4, 2, 4, 8, 6], [ 2, 4, 2, 4, 6, 8], [ 2, 4, 6, 2, 6, 6, 4], [ 2, 4, 2, 4, 8, 6, 4], [ 2, 4, 2, 4, 6, 8, 4], [ 2, 6, 4, 2, 6, 4, 6], [ 2, 4, 6, 2, 6, 4, 6], [ 2, 6, 4, 2, 4, 6, 6], [ 2, 6, 4, 2, 4, 2, 10], [ 2, 4, 2, 4, 6, 2, 10], [ 2, 4, 2, 4, 6, 2, 6, 6, 4, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 6, 6], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 10], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 8]
79: [ 2], [ 4], [ 6], [ 4, 2, 4, 8], [ 2, 4, 2, 10], [ 2, 4, 2, 4, 6, 8], [ 2, 4, 6, 2, 6, 4, 8], [ 2, 6, 4, 2, 4, 6, 8], [ 2, 4, 2, 4, 6, 2, 10], [ 2, 6, 4, 2, 4, 2, 12], [ 2, 4, 2, 4, 6, 2, 12], [ 2, 4, 6, 2, 6, 4, 2, 10], [ 2, 4, 2, 4, 6, 2, 6, 10], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 12], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 8]
83: [ 2], [ 4], [ 2, 6], [ 4, 6], [ 4, 2, 4, 6, 6], [ 2, 4, 2, 4, 8], [ 4, 2, 4, 2, 10], [ 4, 2, 4, 6, 2, 6, 6], [ 4, 2, 4, 2, 4, 8, 6], [ 2, 4, 2, 4, 6, 2, 10], [ 4, 6, 2, 4, 6, 2, 6, 6], [ 4, 2, 4, 6, 6, 2, 6, 6], [ 2, 4, 6, 2, 6, 4, 6, 6], [ 4, 2, 4, 6, 2, 6, 6, 6], [ 2, 4, 2, 4, 6, 2, 10, 6], [ 4, 2, 4, 6, 2, 6, 4, 8], [ 2, 4, 6, 2, 6, 4, 2, 10], [ 2, 4, 2, 4, 6, 2, 6, 10], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 8]
89: [ 2], [ 4], [ 6], [ 2, 4, 8], [ 4, 2, 10], [ 2, 4, 2, 4, 6, 8], [ 4, 2, 4, 6, 2, 10], [ 4, 2, 4, 6, 2, 6, 4, 8], [ 2, 4, 2, 4, 6, 2, 6, 10], [ 4, 2, 4, 6, 6, 2, 6, 4, 8], [ 4, 2, 4, 2, 4, 8, 6, 4, 8], [ 4, 6, 2, 6, 4, 2, 4, 6, 8], [ 2, 4, 6, 2, 6, 4, 6, 2, 10], [ 2, 6, 4, 2, 4, 6, 6, 2, 10], [ 2, 6, 4, 2, 4, 2, 10, 2, 10], [ 2, 4, 2, 4, 6, 2, 10, 2, 10], [ 2, 4, 6, 2, 6, 4, 2, 6, 10], [ 2, 4, 2, 4, 6, 2, 6, 6, 10], [ 4, 6, 2, 6, 4, 2, 4, 2, 12], [ 2, 4, 6, 2, 6, 4, 2, 4, 12], [ 2, 4, 2, 4, 6, 2, 6, 4, 12], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 8]
97: [ 2], [ 4], [ 6], [ 8], [ 2, 10], [ 2, 12], [ 2, 6, 10], [ 2, 4, 12], [ 2, 6, 6, 10], [ 2, 4, 8, 10], [ 2, 6, 4, 12], [ 4, 2, 6, 12], [ 2, 4, 6, 12], [ 4, 2, 4, 14], [ 2, 4, 2, 4, 8, 6, 10], [ 2, 4, 2, 4, 6, 8, 10], [ 4, 2, 4, 6, 2, 6, 12], [ 2, 4, 2, 4, 6, 2, 16], [ 2, 4, 2, 4, 6, 2, 6, 6, 4, 12], [ 2, 6, 4, 2, 4, 6, 6, 2, 6, 12], [ 2, 4, 2, 4, 6, 2, 6, 4, 6, 12], [ 2, 4, 2, 4, 6, 2, 10, 2, 4, 14], [ 2, 4, 2, 4, 6, 2, 6, 6, 4, 14], [ 2, 4, 6, 2, 6, 4, 2, 4, 6, 14], [ 2, 4, 2, 4, 6, 2, 6, 4, 6, 14], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 16], [ 2, 4, 2, 4, 6, 2, 6, 4, 2, 18], [ 2, 4, 6, 2, 6, 4, 2, 4, 6, 8, 10], [ 2, 4, 6, 2, 6, 4, 2, 4, 2, 12, 10], [ 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 12], [ 2, 4, 6, 2, 6, 4, 2, 4, 2, 10, 12]
[/CODE]
Computation time: less than a minute, but I believe it is possible to do it in less than a second with some really optimised code.

Bobby Jacobs 2022-05-22 15:15

Very good! It seems like the popularity of a prime as the start of a maximal gap is based upon the gaps before p. However, there are weird exceptions like 29 and 37. The 2 gaps before 29 are (4, 6) and the gaps before 37 are (2, 6), but 29 is more popular than 37. If (2, 8) was an admissible gap combination, then that would be a forbidden gap combination for 29, but not 37. However, (2, 8) is not admissible.

By the way, I meant to use <= instead of >= in my previous post. The forbidden gap combinations are minimal admissible k-tuples <= the k gaps before p[SUB]m[/SUB]. How do I make the correct symbols for <= and >=?

mart_r 2022-06-02 20:15

1 Attachment(s)
I have finally fully figured out how to tackle the behaviour of [$]o_n(x)[/$] - i.e. the number of occurrences of primes [$]p_n[/$] as initial members of maximal gaps between non-consecutive primes [$]p_n[/$] and [$]p_{n+k}[/$] for all [$]k<=x[/$].

Again, sincere thanks to Bobby for pushing me in the right direction. Although, "forbidden gap constellations" sounds kind of illegal, anyone mind if I call them "blocking patterns" or similar? Suggestions are welcome.

So, let [$]B(p_n)[/$] be the set of blocking patterns for the prime [$]p_n[/$], for example [$]B(31)=\lbrace\lbrace0,2\rbrace,\lbrace0,2,6\rbrace,\lbrace0,2,8\rbrace,\lbrace0,2,8,12\rbrace,\lbrace0,2,6,12\rbrace\rbrace[/$]. (Correspondingly, the blocking gap patterns are [$]\lbrace\lbrace2\rbrace,\lbrace2,4\rbrace,\lbrace2,6\rbrace,\lbrace2,6,4\rbrace,\lbrace2,4,6\rbrace\rbrace[/$].)

These patterns form a minimal set of sorts. I got temporarily addicted to try and find as many of them as possible. Much to my surprise, I recently even managed to get up to p=97 in less than a second even though my code is far from being optimised, but computation time is ballooning exponentially for larger p. The list in the attachment is not guaranteed to be exhaustive.

To evaluate [$]o_n(x)[/$] directly, we subtract from x the number of occurrences of all patterns in [$]B(p_n)[/$] in the range [[$]p_{n-k+2}[/$], [$]p_{n+x}[/$]], where k is the cardinality of the pattern.

By looking at the table of blocking patterns, we can now see right away, for example, that 29 occurs more often than 37 for large x by a margin equivalent to the number of occurrences of the patterns {0,2,6,14} and {0,2,6,8,18} below x. This answers post # 44.

[$]o_n(x)[/$] remains large if there are very few blocking patterns. [$]n=2[/$] has none because [$]B(p_n)=B(3)=\lbrace0,1\rbrace[/$], a non-admissible prime pattern for [$]p>=3[/$], hence [$]o_1(x)=o_2(x)=x[/$].
[$]n=3[/$] and [$]n=4[/$] only have [$]\lbrace0,2\rbrace[/$] as blocking patterns, all larger n have at least one pattern more, thus [$]o_4(x)>o_n(x)[/$] for all [$]n>4[/$] and [$]x>18[/$] (particularly, [$]o_4(x)=x+2-\#(twin\:primes\:below\:p_{4+x})[/$]).
There's [$]n=8[/$] with [$]B(19)=\lbrace\lbrace0,2\rbrace,\lbrace0,2,6\rbrace\rbrace[/$], a minimum for its kind, only twins and the first kind of triplets are blocked, and all [$]n>8[/$] have at least one more blocking pattern, or one that is more common, like [$]\lbrace0,6\rbrace[/$]. We have [$]o_8(x)>o_n(x)[/$] for all [$]n>8[/$] and [$]x>496[/$] (i.e. p=19 "cannot be beaten" above that point).

The asymptotic growth rate of [$]o_n(x)[/$] can be obtained via the blocking patterns with additional regard to the open residue classes in each pattern. For x large, [$]o_{29}(x)[/$] ([$]p_{29}=109[/$]) differs from [$]o_8(x)[/$] only by a margin of the number of occurrences of sextuples [$]\lbrace0,2,8,12,14,20\rbrace[/$], [$]\lbrace0,2,6,12,14,20\rbrace[/$], [$]\lbrace0,2,6,8,12,18\rbrace[/$], and [$]\lbrace0,2,6,8,12,20\rbrace[/$] with a total of 8 open residue classes mod 210; in terms of error this is [$]O(x\cdot(\log x)^{-6})[/$]. We can leave septuples or longer patterns out of the game as the have [$]O(x\cdot(\log x)^{-7})[/$] or smaller. If p=109 should be beaten in the long run, it requires, apart from the minimum of [$]\lbrace0,2\rbrace[/$] and [$]\lbrace0,2,6\rbrace[/$] as blocking patterns, either sextuples with less open residue classes in total, or no sextuples at all. And of course, no quadruple or quintuple blocking pattern as well. The next candidate for this is p=5659: only [$]\lbrace0,2,6,8,12,18\rbrace[/$] gets blocked, and this pattern has only one open residue class mod 210.

Regarding p=5659 vs. p=9439 (my fallacy in post # 40), the latter seems to be in the lead judging by the small numbers because of the millions of possible blocking patterns in favor of p=9439, but these have a cardinality of as small as 5. At [$]x=10^6[/$], p=9439 is in the lead by more than 30,000 - it takes at least as many quintuples of the forms [$]\lbrace0,4,6,12,16\rbrace[/$], [$]\lbrace0,4,6,10,16\rbrace[/$], [$]\lbrace0,6,8,14,18\rbrace[/$], [$]\lbrace0,2,8,14,18\rbrace[/$], [$]\lbrace0,6,10,12,18\rbrace[/$], [$]\lbrace0,4,10,12,18\rbrace[/$], [$]\lbrace0,6,8,12,18\rbrace[/$], [$]\lbrace0,2,8,12,18\rbrace[/$], [$]\lbrace0,2,6,12,18\rbrace[/$], [$]\lbrace0,4,6,10,18\rbrace[/$], or [$]\lbrace0,2,6,8,18\rbrace[/$] until p=5659 can overtake p=9439, we expect this not to happen before [$]x=10^8[/$].

To conclude, the primes for which a local maximum as described in post # 35 is reached for [$]\lim x\to\infty[/$], or rather, for sufficiently large x, should be equal to [$]2\:(3), 7,[/$] and [$]19[/$], with infinitely many [$]o_n(x)[/$] for [$]n>8[/$] coming arbitrarily close to [$]o_8(x)-O(x\cdot(\log x)^{-6})[/$] (e.g. the primes [$]5659[/$] ([$]n=746[/$]), [$]88819[/$] ([$]n=8605[/$]), [$]855739[/$] ([$]n=68032[/$]), [$]74266279[/$] ([$]n=4353833[/$]), [$]964669639[/$] ([$]n=49141276[/$]), [$]9853497769[/$] ([$]n=448687813[/$]), etc. each move toward this upper bound from below). So my previous implicit assumption that the list of primes with local maxima, bounded from above, is infinite was wrong.


Phew, that took long enough. And it's only framework, sort of. Also poorly worded at times, but I really need to finish this off now, one way or another.
I flip out if now someone gives me a link to some obscure 19th century work that covers all this...

mart_r 2022-06-09 17:26

"Sitting target / sitting, waiting / anticipating / nothing / nothing."
 
CSG looks well-behaved even for k <= 1024 (p in range < 10^14):

[CODE] k gap CSG_max p
112 5940 1.05550107 36683716323847
116 6052 1.02516052 36683716323619
120 6220 1.03269957 36683716323283
124 6388 1.04043735 36683716323283
128 6510 1.01858817 36683716323161
132 6642 1.00390061 36683716323167
136 6742 0.96976743 36683716323109
140 6658 0.94384648 17674627574311
144 6840 0.96488178 17674627574369
148 6992 0.96688912 17674627574141
152 7126 0.95779790 17674627574083
156 7460 0.97452838 30512335335437
160 7614 0.97643675 30512335335437
164 7732 0.95708911 30512335335319
168 7946 0.99499058 30512335334951
172 8100 0.99726110 30512335334797
176 8254 0.99967015 30512335334797
180 8364 0.97661645 30512335335299
184 8510 0.97483498 30512335335059
188 8736 1.01921576 30512335334927
192 8892 1.02319197 30512335334771
196 9004 1.00215904 30512335334797
200 9148 1.01448166 28330683392731
204 9324 1.03039259 28330683392659
208 9492 1.04177291 28330683392597
212 9630 1.03626675 28330683392353
216 9778 1.03654152 28330683392371
220 9856 0.99828866 28330683392371
224 9974 0.98269325 28330683392147
228 10058 0.94929275 28330683392129
232 8294 0.94835143 185067241757
236 9700 0.95641246 5185992136441
240 9850 0.96394780 5185992136453
244 10626 0.94205155 28330683392597
248 10818 0.96644566 28330683392371
252 10596 0.94771341 12666866223047
256 11310 0.93908073 52248744686339
260 11476 0.94818065 52248744686197
264 11604 0.93866201 52248744686069
268 11724 0.92547977 52248744686197
272 11264 0.93001522 12666866223047
276 12106 0.91167574 68182243872601
280 11752 0.91251084 21947823205027
284 11920 0.92535164 21947823205027
288 12096 0.94216306 21947823204943
292 12310 0.94178965 25698372297889
296 12460 0.94533825 25698372297691
300 12704 0.99505355 25698372297029
304 12920 1.03124573 25698372297029
308 13170 1.08475215 25698372297029
312 13308 1.08194976 25698372297029
316 13482 1.09729159 25698372297029
320 13616 1.09257083 25698372296963
324 13728 1.07704353 25698372296873
328 13878 1.08048911 25698372297029
332 13986 1.06336781 25698372297007
336 14136 1.06693878 25698372296963
340 14234 1.04538481 25698372296873
344 14336 1.02612587 25698372296243
348 14466 1.02043217 25698372295733
352 14642 1.03657564 25698372297029
356 14778 1.03379163 25698372295733
360 14890 1.01983645 25698372295711
364 15044 1.02563857 25698372296963
368 15222 1.04265118 25698372295019
372 15360 1.04099593 25698372294839
376 15546 1.06172759 25698372295033
380 15694 1.06474083 25698372294457
384 15832 1.06313720 25698372294409
388 15968 1.06066576 25698372294611
392 16158 1.08316800 25698372294421
396 16242 1.05682221 25698372294337
400 16344 1.03908815 25698372294563
404 16536 1.06228245 25698372294457
408 16678 1.06277691 25698372294421
412 16762 1.03722225 25698372294337
416 16852 1.01477000 25698372294457
420 16974 1.00672310 25698372295033
424 17160 1.02688378 25698372294421
428 17302 1.02766540 25698372294421
432 17396 1.00751345 25698372294611
436 17586 1.02931311 25698372294421
440 17724 1.02843581 25698372294409
444 17810 1.00513477 25698372294323
448 17886 0.97797632 25698372294253
452 17972 0.95549900 25698372294281
456 18114 0.95669262 25698372293557
460 18234 0.94875594 25698372293809
464 18390 0.95581939 25698372293597
468 18536 0.95873888 25698372293597
472 19506 0.94029942 93152147737543
476 19770 0.98628553 93152147737279
480 19878 0.97192308 93152147737199
484 19954 0.94554667 93152147737237
488 19192 0.94334862 25698372294421
492 20322 0.97552201 93152147736727
496 20490 0.98440496 93152147736559
500 20598 0.97040570 93152147736451
504 20748 0.97245252 93152147736301
508 20850 0.95643446 93152147736199
512 21004 0.96004428 93152147736073
516 21260 1.00210684 93152147735789
520 21390 0.99658464 93152147735659
524 21478 0.97538922 93152147735599
528 21592 0.96413821 93152147735371
532 21726 0.96039757 93152147735351
536 21874 0.96185682 93152147735203
540 21964 0.94210589 93152147735113
544 22076 0.93058797 93152147734973
548 22224 0.93216222 93152147733739
552 22486 0.97513465 93152147732647
556 22628 0.97445048 93152147734421
560 22792 0.98180763 93152147734171
564 22958 0.98989507 93152147734091
568 23130 1.00017583 93152147733919
572 23346 1.02661845 93152147733703
576 23524 1.03914014 93152147733553
580 23610 1.01786364 93152147733467
584 23706 1.00050490 93152147733553
588 23912 1.02309904 93152147733137
592 24068 1.02754489 93152147732981
596 24240 1.03781136 93152147732723
598 24402 1.07077256 93152147732647
600 24436 1.05684370 93152147732641
604 24540 1.04234724 93152147732509
608 24676 1.03956218 93152147732401
612 24798 1.03177448 93152147732251
616 24880 1.00980556 93152147732197
620 25008 1.00437652 93152147732069
624 25164 1.00888491 93152147731913
628 25264 0.99368357 93152147731813
632 25348 0.97310855 93152147731729
636 25500 0.97626664 93152147731549
640 25578 0.95395507 93152147731499
644 25696 0.94554893 93152147731381
648 25860 0.95284382 93152147731217
652 26004 0.95333346 93152147731073
656 26252 0.98935642 93152147730797
660 26412 0.99528904 93152147730637
664 26606 1.01294463 93152147730443
668 26706 0.99822651 93152147730371
672 26826 0.99048613 93152147730223
676 26938 0.98010919 93152147730139
680 27094 0.98468729 93152147729983
684 27186 0.96770698 93152147729891
688 27276 0.95027202 93152147729983
692 27368 0.93371537 93152147729891
696 27516 0.93569220 93152147729561
700 27582 0.91092486 93152147729467
704 27698 0.90261024 93152147729561
708 27820 0.89629956 93152147729143
712 27948 0.89196235 93152147729143
716 28048 0.87877867 93152147729143
720 27710 0.89275685 54116590394771
724 27860 0.89636501 54116590394621
728 27998 0.89606095 54116590394483
732 28172 0.90750139 54116590393157
736 28332 0.91437757 54116590394149
740 28536 0.93570243 54116590393991
744 28666 0.93272540 54116590393861
748 28800 0.93108486 54116590393777
752 28982 0.94518150 54116590393499
756 29130 0.94812537 54116590393447
760 29370 0.98143984 54116590393157
764 29456 0.96387023 54116590393121
768 29630 0.97537452 54116590392947
772 29706 0.95469068 54116590393157
776 29826 0.94853391 54116590392947
780 29964 0.94825668 54116590392929
784 30076 0.93959664 54116590392451
788 30192 0.93229670 54116590392947
792 30288 0.91869099 54116590392289
796 30456 0.92805971 54116590392121
800 30654 0.94703768 54116590391873
804 30740 0.93024784 54116590391837
808 30816 0.91051339 54116590391861
812 30990 0.92171618 54116590391873
816 31176 0.93673527 54116590391351
820 31368 0.95371131 54116590391113
824 31516 0.95671400 54116590391011
828 31596 0.93820121 54116590391011
832 31734 0.93807554 54116590391113
836 31852 0.93169439 54116590391011
840 31936 0.91480473 54116590391077
844 32062 0.91104489 54116590391077
848 32158 0.89810577 54116590391011
852 32880 0.89162466 93152147732647
856 33006 0.88736277 93152147732641
860 32594 0.90483328 54116590389887
864 32714 0.89936117 54116590389863
868 32790 0.88065023 54116590389887
872 32960 0.89035364 54116590389887
876 33068 0.88139835 54116590389473
880 33158 0.86716391 54116590389419
884 33276 0.86135519 54116590389863
888 33420 0.86326354 54116590389473
892 33550 0.86104325 54116590388977
896 33738 0.87595633 54116590388789
900 34052 0.86261430 65480290959731
904 34264 0.88413463 65480290959547
908 34380 0.87757423 65480290959403
912 34474 0.86468039 65480290959403
916 35030 0.86491644 93152147730443
920 35156 0.86099754 93152147730497
924 34932 0.87719975 65480290958651
928 35160 0.90333082 65480290958651
932 35254 0.89038566 65480290958557
936 35504 0.89448345 70981263873617
940 35646 0.89544859 70981263873617
944 35654 0.88610601 65480290958129
948 35814 0.89239499 65480290957997
952 36080 0.90067205 70981263873617
956 36204 0.89647169 70981263873617
960 36294 0.88259124 70981263872969
964 32722 0.87857798 3529553758999
968 36550 0.87667128 70981263873109
972 36728 0.88790229 70981263872969
976 36864 0.88721696 70981263872257
980 37006 0.88823910 70981263872257
984 37086 0.87179868 70981263872257
988 37224 0.87172780 70981263872257
992 37440 0.89358331 70981263872257
996 37564 0.88954266 70981263872257
1000 36346 0.87913969 25264345114117
1004 36534 0.89520659 25264345113919
1008 36604 0.87646499 25264345113919
1012 36740 0.87719292 25264345113713
1016 36876 0.87792446 25264345113613
1020 37294 0.87603752 31618998499597
1024 37074 0.85792826 25264345113613
[/CODE]
Update on blocking patterns (see previous post):
p = 157: 1195 patterns on my watch
p = 163: at least 2125 patterns
p = 167: at least 4000 patterns
p = 173: at least 5733 patterns
p = 179: at least 7357 patterns
p = 181: at least 16345 patterns
p = 191: at least 11710 patterns
But the number of patterns is not terribly important (and probably impossible to compute in full for p > 179 or 181) - for some decent comparisons between values of o[SUB]n[/SUB](x), it should be sufficient to know the patterns with cardinality <= 7 or 8 or thereabouts, these are not too hard to figure out if p is not too large.

This is getting boring, I'm going to watch some episodes of PJ Masks now.:popcorn:

Bobby Jacobs 2022-06-12 18:13

[QUOTE=Bobby Jacobs;604770]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.[/QUOTE]

I believe that 223 will eventually beat 127. It starts out behind because there are two consecutive gaps of 12 before 223. However, 223 will eventually catch up with 127 because 127 has a gap of 14. Therefore, the sequence of record lows will start 2, 5, 11, 29, 37, 59, 97, 127, 307, 541, 907, 1151, 1361, ...

mart_r 2022-06-13 18:00

[QUOTE=Bobby Jacobs;607677]I believe that 223 will eventually beat 127. It starts out behind because there are two consecutive gaps of 12 before 223. However, 223 will eventually catch up with 127 because 127 has a gap of 14. Therefore, the sequence of record lows will start 2, 5, 11, 29, 37, 59, 97, 127, 307, 541, 907, 1151, 1361, ...[/QUOTE]

That's what I would assume as well. These would be the primes at the end of a maximal gap, including ones where there is a tie to the previous maximal gap, if the blocking patterns cover more common patterns.

My search is still running, slowly approaching 2e14 for k <= 109.
Does anyone think CSG > 1.1 is possible to find?

mart_r 2022-07-14 20:22

Don't answer me, don't break the silence, don't let me win
 
1 Attachment(s)
Attached are the numbers of first occurrence gaps for k <= 109 and p <= 2e14.
Don't look for me, I'm already moving on.:digging:


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