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 2021-12-02, 18:17 #1 robert44444uk     Jun 2003 Oxford, UK 2,017 Posts Long ranges of gaps larger than a given value Take a range of integers p1 to p(n+1) within which there are primes p1,p2,p3...p(n+1), with gaps between these primes of g1,g2,g3...gn Take an integer to represent the smallest gap in the set g1,g2,g3..gn. Let this smallest gap be x What is the smallest p1 for which x=2,4,6,8,... for each n? If x=4, n=3, then the p1 is 43, as 43,47,53,59 is the first instance of three gaps 4,6,6 that are of size 4 or more The table below shows values at n 1..20 in column 1, then for various x in the other columns. The table values are p1. For x = 1,2 the p1 are 2 and 3 respectively. For clarity I also attach a photo of the table. Sorry if this duplicates other's work, but I have not seen any lists like this in OEIS. Code:  4 6 8 10 12 14 16 18 20 30 50 100 200 1 7 23 89 113 113 113 523 523 887 1327 19609 370261 20831323 2 19 47 199 199 199 1831 2161 2161 3947 24251 413299 72546143 15318488071 3 43 241 683 773 3947 7351 7351 7351 18593 69557 3021173 1284352451 4 73 241 683 7043 10181 23371 63913 66191 112403 668243 14784911 24964088161 5 73 523 683 13477 23087 47161 68281 68281 250501 2352289 107282507 82504446809 6 349 677 1789 13477 23371 47161 231109 231109 783877 4720943 837406403 438290428309 7 349 677 13469 13477 23371 339341 339341 539899 2352247 30097883 2547259079 8 349 677 13469 13477 23371 339841 339841 539899 2722561 64740587 7623725903 9 349 2861 13469 13477 23371 611561 876853 5556769 5556769 130925771 47787139219 10 349 10733 13469 115153 257503 611561 6463309 6463309 7063817 1075978859 96622630009 11 349 10733 62687 202409 406883 611561 7063817 7063817 7063817 1415929063 177013585123 12 661 13421 62687 303731 406883 876853 7591729 12199571 61501877 2695118249 1324527983897 13 661 13421 62687 303731 729979 876853 15022753 15022753 148486951 3731002741 14 661 13421 62687 406817 729979 9355501 15022753 15022753 254259631 7780262561 15 661 13421 200597 406817 729979 12877433 20917291 20917291 562725353 33705791429 16 661 13421 200597 406817 729979 12877433 72733109 89085481 1143902897 46318910903 17 661 13421 568441 913639 4656469 12877433 134798647 180820951 1462517893 103947089669 18 661 13421 568441 913639 4656469 12877433 195503851 195503851 2158109033 103947089669 19 661 13421 568441 970447 4656469 12877433 232253071 381906059 4527970103 184722051989 20 8629 13421 568441 970447 8962013 112843903 232253071 381906059 9156493673 184722051989 Attached Thumbnails   Last fiddled with by robert44444uk on 2021-12-04 at 13:31
 2021-12-03, 12:06 #2 robert44444uk     Jun 2003 Oxford, UK 2,017 Posts The first instance of 7 consecutive 100+ gaps is 5183073661943 with gaps 116,130,114,158,286,102,102 The first instance of 3 consecutive 200+ gaps is provided by 20222645954633 with gaps of 204,216,218 Last fiddled with by robert44444uk on 2021-12-03 at 12:09
 2021-12-04, 08:10 #3 robert44444uk     Jun 2003 Oxford, UK 111111000012 Posts The first instance of 13 and 14 consecutive 50+ gaps is at 3057601284499 where subsequent gaps are 60, 54, 58, 50, 100, 90, 62, 156, 124, 60, 60, 68, 76 and 86
 2021-12-04, 10:47 #4 mart_r     Dec 2008 you know...around... 7·101 Posts A nice variation. The idea of e.g. consecutive gaps with certain merit value went through my head before, but I never got around to calculate specific numbers. I think that would fall out as a corollary of your table. I would suggest continuing this search. You also see, for example, the gaps in twin primes depicted by long runs of the same value in the column x=4.
 2021-12-04, 14:38 #5 Dr Sardonicus     Feb 2017 Nowhere 2×2,687 Posts I note that for a given x, the sequence of least primes with all g's $\ge$ x for increasing n (down a given column of the table) is monotone increasing; likewise for a given n and increasing lower bound x on gap size (along a given row). The repeated values along the rows and columns are interesting - there are "jumps" in the gap size for given n, and in the number of consecutive primes with given minimum gap size. For n = 1, the gap size has to be even to for (p, p + g1) to be an "admissible" pair. For n+1 > 2 consecutive primes, the gap values are further constrained by the n + 1-tuple (p, p + g1, p + g1 + g2, ...) being "admissible." In particular, if "x" is not divisible by 3, the triple (p, p + x, p + 2*x) is not admissible (one of them is divisible by 3 for p > 3). I suppose one could look at n-tuples of gaps (g1, g2,... gn) with all g's $\ge$ x, (but not much larger) for which (p, p + g1, p + g1 + g2, ... p + g1 + ... + gn) is admissible. One can probably use the Hardy-Littlewood conjectural asymptotic formula to estimate how large p has to be for there to be a decent chance of the tuple to consist entirely of primes; I don't know about the condition that they be consecutive primes.
 2021-12-05, 22:11 #6 Bobby Jacobs     May 2018 5·72 Posts This is very cool!
 2021-12-06, 10:16 #7 robert44444uk     Jun 2003 Oxford, UK 7E116 Posts I'm taking all gaps up to 50 in case anyone is interested in extending other ranges. 60 looks a soft target, as are all 0mod30 cases What I find gratifying about this search is that the higher you go, you do find new records. The small gap cases are very easy to extend. For example - 4 Code: 20 8629 21 8629 22 8629 23 8629 24 13399 25 13399 26 13399 27 14629 28 14629 29 24421 30 24421 31 24421 32 24421 33 24421 34 24421 35 24421 36 24421 37 24421 38 24421 39 24421 40 24421 41 24421 42 62299 43 62299 44 62299 45 62299 46 62299 47 62299 48 62299 49 62299 50 62299 51 62299 52 62299 53 62299 54 187909 55 187909 56 187909 57 187909 58 187909 59 187909 60 187909 61 187909 62 187909 63 187909 64 187909 65 187909 66 187909 67 187909 68 187909 69 187909 70 187909 71 187909
2021-12-13, 16:24   #8
robert44444uk

Jun 2003
Oxford, UK

2,017 Posts

Quote:
 Originally Posted by mart_r A nice variation. The idea of e.g. consecutive gaps with certain merit value went through my head before, but I never got around to calculate specific numbers. I think that would fall out as a corollary of your table.
Just out of interest, I calculated the first few ranges of consecutive gaps, each with merit >1. The starting primes are not on OEIS

Code:
Min	merit:	1	Run	of	merits:	1	Initial	prime:	2	Following	gaps:	1	Merits:	1.44
Min	merit:	1	Run	of	merits:	2	Initial	prime:	2	Following	gaps:	1	2	Merits:	1.44	1.820478453
Min	merit:	1	Run	of	merits:	3	Initial	prime:	2	Following	gaps:	1	2	2	Merits:	1.44	1.820478453	1.242669869
Min	merit:	1	Run	of	merits:	4	Initial	prime:	2	Following	gaps:	1	2	2	4	Merits:	1.44	1.820478453	1.242669869	2.055593369
Min	merit:	1	Run	of	merits:	5	Initial	prime:	683	Following	gaps:	8	10	8	10	8	Merits:	1.225772819	1.529487021	1.220906581	1.523494836	1.216200763
Min	merit:	1	Run	of	merits:	6	Initial	prime:	1789	Following	gaps:	12	10	12	8	16	14	Merits:	1.602261949	1.3340275	1.599651392	1.065496216	2.129750371	1.861375897
Min	merit:	1	Run	of	merits:	7	Initial	prime:	13477	Following	gaps:	10	12	14	10	14	16	14	Merits:	1.051664069	1.261898449	1.472077185	1.05136911	1.471802284	1.681876797	1.471459487
Min	merit:	1	Run	of	merits:	8	Initial	prime:	13477	Following	gaps:	10	12	14	10	14	16	14	10	Merits:	1.051664069	1.261898449	1.472077185	1.05136911	1.471802284	1.681876797	1.471459487	1.05092845
Min	merit:	1	Run	of	merits:	9	Initial	prime:	13477	Following	gaps:	10	12	14	10	14	16	14	10	14	Merits:	1.051664069	1.261898449	1.472077185	1.05136911	1.471802284	1.681876797	1.471459487	1.05092845	1.47118591
Min	merit:	1	Run	of	merits:	10	Initial	prime:	611561	Following	gaps:	26	16	18	20	16	14	22	14	22	24	Merits:	1.951399644	1.200857488	1.350962021	1.501065596	1.20084953	1.050741276	1.651162026	1.050736635	1.651154732	1.801254846
Min	merit:	1	Run	of	merits:	11	Initial	prime:	611561	Following	gaps:	26	16	18	20	16	14	22	14	22	24	38	Merits:	1.951399644	1.200857488	1.350962021	1.501065596	1.20084953	1.050741276	1.651162026	1.050736635	1.651154732	1.801254846	2.851978442
Min	merit:	1	Run	of	merits:	12	Initial	prime:	876853	Following	gaps:	18	22	20	16	18	24	32	24	16	14	16	18	Merits:	1.315395755	1.607703511	1.461545967	1.169234825	1.315387424	1.753847268	2.338458347	1.753839084	1.169223717	1.023069389	1.169220795	1.315371641
Min	merit:	1	Run	of	merits:	13	Initial	prime:	876853	Following	gaps:	18	22	20	16	18	24	32	24	16	14	16	18	18	Merits:	1.315395755	1.607703511	1.461545967	1.169234825	1.315387424	1.753847268	2.338458347	1.753839084	1.169223717	1.023069389	1.169220795	1.315371641	1.315369668
Min	merit:	1	Run	of	merits:	14	Initial	prime:	15022753	Following	gaps:	18	22	18	20	22	20	40	24	26	28	36	32	22	56	Merits:	1.089253658	1.33130993	1.089253482	1.210281559	1.331309608	1.210281355	2.420562514	1.452337275	1.573365229	1.694393146	2.178505227	1.93644881	1.331308385	3.388784681
Min	merit:	1	Run	of	merits:	15	Initial	prime:	20917291	Following	gaps:	22	18	18	18	42	32	18	24	34	36	44	22	24	18	42	Merits:	1.305166519	1.067863449	1.067863395	1.06786334	2.491681	1.898423393	1.067863061	1.423817343	2.017074431	2.135725662	2.610331099	1.305165387	1.423816696	1.06786245	2.491678922

2021-12-14, 17:33   #9
robert44444uk

Jun 2003
Oxford, UK

2,017 Posts

Quote:
 Originally Posted by robert44444uk Just out of interest, I calculated the first few ranges of consecutive gaps, each with merit >1. The starting primes are not on OEIS Code: Min merit: 1 Run of merits: 1 Initial prime: 2 Following gaps: 1 Merits: 1.44 Min merit: 1 Run of merits: 2 Initial prime: 2 Following gaps: 1 2 Merits: 1.44 1.820478453 Min merit: 1 Run of merits: 3 Initial prime: 2 Following gaps: 1 2 2 Merits: 1.44 1.820478453 1.242669869 Min merit: 1 Run of merits: 4 Initial prime: 2 Following gaps: 1 2 2 4 Merits: 1.44 1.820478453 1.242669869 2.055593369 Min merit: 1 Run of merits: 5 Initial prime: 683 Following gaps: 8 10 8 10 8 Merits: 1.225772819 1.529487021 1.220906581 1.523494836 1.216200763 Min merit: 1 Run of merits: 6 Initial prime: 1789 Following gaps: 12 10 12 8 16 14 Merits: 1.602261949 1.3340275 1.599651392 1.065496216 2.129750371 1.861375897 Min merit: 1 Run of merits: 7 Initial prime: 13477 Following gaps: 10 12 14 10 14 16 14 Merits: 1.051664069 1.261898449 1.472077185 1.05136911 1.471802284 1.681876797 1.471459487 Min merit: 1 Run of merits: 8 Initial prime: 13477 Following gaps: 10 12 14 10 14 16 14 10 Merits: 1.051664069 1.261898449 1.472077185 1.05136911 1.471802284 1.681876797 1.471459487 1.05092845 Min merit: 1 Run of merits: 9 Initial prime: 13477 Following gaps: 10 12 14 10 14 16 14 10 14 Merits: 1.051664069 1.261898449 1.472077185 1.05136911 1.471802284 1.681876797 1.471459487 1.05092845 1.47118591 Min merit: 1 Run of merits: 10 Initial prime: 611561 Following gaps: 26 16 18 20 16 14 22 14 22 24 Merits: 1.951399644 1.200857488 1.350962021 1.501065596 1.20084953 1.050741276 1.651162026 1.050736635 1.651154732 1.801254846 Min merit: 1 Run of merits: 11 Initial prime: 611561 Following gaps: 26 16 18 20 16 14 22 14 22 24 38 Merits: 1.951399644 1.200857488 1.350962021 1.501065596 1.20084953 1.050741276 1.651162026 1.050736635 1.651154732 1.801254846 2.851978442 Min merit: 1 Run of merits: 12 Initial prime: 876853 Following gaps: 18 22 20 16 18 24 32 24 16 14 16 18 Merits: 1.315395755 1.607703511 1.461545967 1.169234825 1.315387424 1.753847268 2.338458347 1.753839084 1.169223717 1.023069389 1.169220795 1.315371641 Min merit: 1 Run of merits: 13 Initial prime: 876853 Following gaps: 18 22 20 16 18 24 32 24 16 14 16 18 18 Merits: 1.315395755 1.607703511 1.461545967 1.169234825 1.315387424 1.753847268 2.338458347 1.753839084 1.169223717 1.023069389 1.169220795 1.315371641 1.315369668 Min merit: 1 Run of merits: 14 Initial prime: 15022753 Following gaps: 18 22 18 20 22 20 40 24 26 28 36 32 22 56 Merits: 1.089253658 1.33130993 1.089253482 1.210281559 1.331309608 1.210281355 2.420562514 1.452337275 1.573365229 1.694393146 2.178505227 1.93644881 1.331308385 3.388784681 Min merit: 1 Run of merits: 15 Initial prime: 20917291 Following gaps: 22 18 18 18 42 32 18 24 34 36 44 22 24 18 42 Merits: 1.305166519 1.067863449 1.067863395 1.06786334 2.491681 1.898423393 1.067863061 1.423817343 2.017074431 2.135725662 2.610331099 1.305165387 1.423816696 1.06786245 2.491678922
Checked up to p=1001200549 and a second run of 15 found but no run of 16 yet.

Min merit: 1 Run of merits: 15 Initial prime: 34398523 Following gaps: 40 56 40 18 26 36 18 22 20 18 34 56 22 24 24 Merits: 2.30500730414805 3.22701000956899 2.3050069334542 1.03725305054945 1.49825436117103 2.07450594818822 1.03725291153983 1.2677535203211 1.15250315781684 1.03725280728285 1.95925524356651 3.22700845266168 1.26775320175876 1.38300344185791 1.3830033862545

Because the median merit following a prime is only about 0.7 merit - i.e. 0.7*g/ln(p) - there may be a case for saying that arbitrarily long runs of gaps with merit >1 do not occur. Interest to hear from others.

Last fiddled with by robert44444uk on 2021-12-14 at 17:39

 2021-12-16, 09:02 #10 robert44444uk     Jun 2003 Oxford, UK 2,017 Posts I found the first instance of two consecutive 300 gaps. 2 ||| 1362810282139 1362810282439 1362810282781 || 300 342 In the search I discovered 6 such. The largest gap following the 300 pair found was: 2 ||| 1724709287369 1724709287671 1724709287999 || 302 328 80
 2021-12-20, 14:54 #11 robert44444uk     Jun 2003 Oxford, UK 2,017 Posts some findings about strings of minimum merits: Merit >2 Code: Min merit: 2 Run of merits: 2 Initial prime: 199 Following gaps: 12 12 Merits: 2.267014728 2.242211901 Min merit: 2 Run of merits: 3 Initial prime: 7351 Following gaps: 18 24 18 Merits: 2.021883147 2.695103818 2.020590063 Min merit: 2 Run of merits: 4 Initial prime: 648449 Following gaps: 32 28 54 44 Merits: 2.391211347 2.092302213 4.03514125 3.287872413 Min merit: 2 Run of merits: 5 Initial prime: 2352323 Following gaps: 34 30 50 30 40 Merits: 2.317512986 2.044862385 3.408101012 2.044857645 2.726474489 Min merit: 2 Run of merits: 6 Initial prime: 30097967 Following gaps: 54 42 54 66 42 60 Merits: 3.135894622 2.439028896 3.135894041 3.832758984 2.439028134 3.484325623 Min merit: 2 Run of merits: 7 Initial prime: 335739713 Following gaps: 40 84 42 44 40 56 40 Merits: 2.03750573 4.278762006 2.139380976 2.241256246 2.037505665 2.852507913 2.037505635 Min merit: 2 Run of merits: 8 Initial prime: 1065726007 Following gaps: 42 48 52 54 48 50 76 50 Merits: 2.02050115 2.309144167 2.501572842 2.597787176 2.30914415 2.405358485 3.656144889 2.405358471 Merit >3 Code: Min merit: 3 Run of merits: 2 Initial prime: 38461 Following gaps: 40 42 Merits: 3.788811632 3.977860556 Min merit: 3 Run of merits: 3 Initial prime: 740749 Following gaps: 52 48 42 Merits: 3.847458022 3.551481267 3.107531211 Min merit: 3 Run of merits: 4 Initial prime: 14784977 Following gaps: 66 50 52 60 Merits: 3.997790871 3.028628629 3.149773129 3.634352836 Min merit: 3 Run of merits: 5 Initial prime: 754473523 Following gaps: 126 88 62 84 66 Merits: 6.163921998 4.304961361 3.033040941 4.109281259 3.228720971 Merit >4 Code: Min merit: 4 Run of merits: 2 Initial prime: 413299 Following gaps: 54 58 Merits: 4.175711922 4.484978606 Min merit: 4 Run of merits: 3 Initial prime: 112219777 Following gaps: 78 106 114 Merits: 4.208034635 5.718610957 6.150203923 Min merit: 4 Run of merits: 4 Initial prime: 5289589577 Following gaps: 180 92 90 100 Merits: 8.039660002 4.10915955 4.019829992 4.466477765 Min merit >5 Code: Min merit: 5 Run of merits: 2 Initial prime: 10938023 Following gaps: 102 96 Merits: 6.293287056 5.923090292 Min merit: 5 Run of merits: 3 Initial prime: 1440754039 Following gaps: 122 130 108 Merits: 5.785162115 6.164516983 5.12129101

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