Gaps between non-consecutive primes - Page 5

Bobby Jacobs · 2022-04-29, 23:22

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]]

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:

Originally Posted by Bobby Jacobs

Here are the forbidden gap combinations of the first few primes.

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_m be the mth prime. Suppose the k gaps before the (m+n)th prime are one of these forbidden k-tuples. If p_m+n-p_m+n-k<=p_m-p_m-k, then p_m+n-k-p_m-k>=p_m+n-p_m. 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_m.

mart_r · 2022-05-17, 17:02

CSG_max for p<=10¹⁴:

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

And just above 10¹⁴, 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

@ 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]

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_m. How do I make the correct symbols for <= and >=?

mart_r · 2022-06-02, 20:15

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

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

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_n(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.

Bobby Jacobs · 2022-06-12, 18:13

Quote:

Originally Posted by 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.

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:

Originally Posted by Bobby Jacobs

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, ...

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

Attached are the numbers of first occurrence gaps for k <= 109 and p <= 2e14.
Don't look for me, I'm already moving on.

2022-04-29, 23:22	#45
Bobby Jacobs May 2018 2³×5×7 Posts	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]] 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.

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2022-05-11, 16:28	#47
Bobby Jacobs May 2018 2³×5×7 Posts	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_m be the mth prime. Suppose the k gaps before the (m+n)th prime are one of these forbidden k-tuples. If p_m+n-p_m+n-k<=p_m-p_m-k, then p_m+n-k-p_m-k>=p_m+n-p_m. 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_m.

2022-05-22, 15:15	#50
Bobby Jacobs May 2018 2³×5×7 Posts	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_m. How do I make the correct symbols for <= and >=?