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 2022-04-29, 23:22 #45 Bobby Jacobs     May 2018 32·29 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.
2022-05-01, 14:56   #46
mart_r

Dec 2008
you know...around...

3×11×23 Posts

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?

 2022-05-11, 16:28 #47 Bobby Jacobs     May 2018 32×29 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 pm be the mth prime. Suppose the k gaps before the (m+n)th prime are one of these forbidden k-tuples. If pm+n-pm+n-k<=pm-pm-k, then pm+n-k-pm-k>=pm+n-pm. 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 pm.
 2022-05-17, 17:02 #48 mart_r     Dec 2008 you know...around... 3·11·23 Posts CSGmax 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).
 2022-05-19, 17:28 #49 mart_r     Dec 2008 you know...around... 3×11×23 Posts 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.
 2022-05-22, 15:15 #50 Bobby Jacobs     May 2018 32·29 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 pm. How do I make the correct symbols for <= and >=?
2022-06-02, 20:15   #51
mart_r

Dec 2008
you know...around...

13678 Posts

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...
Attached Files
 blocking_gap_patterns.txt (80.5 KB, 10 views)

 2022-06-09, 17:26 #52 mart_r     Dec 2008 you know...around... 3×11×23 Posts "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 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 on(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.
2022-06-12, 18:13   #53
Bobby Jacobs

May 2018

32·29 Posts

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, ...

2022-06-13, 18:00   #54
mart_r

Dec 2008
you know...around...

2F716 Posts

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?

Last fiddled with by mart_r on 2022-06-13 at 18:04

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