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Old 2019-04-18, 07:36   #12
Thomas11
 
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Feb 2003

22·32·53 Posts
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Here are the extensions to ATH's results up to k=1000B (=1e12):

Consecutive first occurrence gaps 1359 up to 1635, and nonconsecutive first occurrence gaps up to 2377.

The extensions to the maximal gap list are given below.

Code:
gap	k
1359	238021889388
1360	120701590522
1361	103656076897
1362	129780127055
1363	45464706810
1364	236116004983
1365	27432631065
1366	221323766702
1367	66664704428
1368	79675959260
1369	105795508798
1370	22751684835
1371	150422884902
1372	73987959848
1373	91101657665
1374	74619205678
1375	106433204497
1376	109109662112
1377	140926519020
1378	158222902962
1379	59513006413
1380	83535481318
1381	81120569787
1382	208297454633
1383	82037154155
1384	91105669713
1385	35954269187
1386	36654125322
1387	82436707565
1388	52106714982
1389	335699670478
1390	134783026287
1391	91527303572
1392	160241217295
1393	78441568602
1394	107218832948
1395	93204583073
1396	370369427962
1397	71529209675
1398	89717150497
1399	167763335213
1400	78769799252
1401	113406055667
1402	83591984703
1403	209318000600
1404	115946907303
1405	60066856568
1406	157757582267
1407	207000817860
1408	75688827280
1409	131711929323
1410	199574686315
1411	48278160327
1412	157757709578
1413	32135929095
1414	179075794238
1415	49037845735
1416	64167955752
1417	117577979190
1418	108513597027
1419	77078300443
1420	74951308542
1421	42119304362
1422	101735004478
1423	142152878497
1424	258224787443
1425	88942037140
1426	114261540757
1427	228073106628
1428	100389566412
1429	175835907578
1430	48978765967
1431	105371512887
1432	89851341625
1433	211777215930
1434	197709087693
1435	110562730142
1436	34440671017
1437	141185526520
1438	167850568830
1439	94595234198
1440	34516209527
1441	195969981917
1442	136888972125
1443	134581666552
1444	159728123338
1445	38840382823
1446	227253338462
1447	57301290370
1448	77920988750
1449	102332614223
1450	108516370937
1451	64095115322
1452	151527991675
1453	145272707550
1454	174799142838
1455	81631078743
1456	195778814802
1457	301410421995
1458	264657155665
1459	206268027728
1460	64584093742
1461	472019855132
1462	88475830910
1463	69741598787
1464	224994765053
1465	135767046975
1466	275157656942
1467	85661945075
1468	63139736062
1469	228677692513
1470	106208976653
1471	156171984272
1472	138299223690
1473	184783689402
1474	365110475063
1475	67410836280
1476	267918092432
1477	185821439738
1478	96139203857
1479	214041248743
1480	240784621312
1481	228476499377
1482	176427443750
1483	48773094382
1484	136319467498
1485	169809420105
1486	150516739082
1487	405178602348
1488	209618644275
1489	130741136258
1490	88340661855
1491	227858144272
1492	121639858400
1493	66924164457
1494	406142933448
1495	69782047618
1496	174973906777
1497	291869230735
1498	308394175852
1499	22339570278
1500	172823544343
1501	326074608547
1502	128837925765
1503	181674927455
1504	289506845778
1505	160823433270
1506	227491883602
1507	176020578323
1508	279911847660
1509	434412984948
1510	163775464003
1511	393469552312
1512	218349176503
1513	342721571537
1514	440909475828
1515	155390806498
1516	85283577452
1517	196413282468
1518	184159612907
1519	235557862248
1520	159460255028
1521	359489269902
1522	219322573465
1523	147488057610
1524	263159931988
1525	98616660978
1526	56105289977
1527	194013931730
1528	263122426457
1529	137914593968
1530	164942161335
1531	82643395422
1532	238243169945
1533	247333123902
1534	450299367998
1535	328710671530
1536	151913191462
1537	343278293043
1538	119496454527
1539	225373027793
1540	139666887840
1541	251898401212
1542	181392272663
1543	180056667807
1544	146578135928
1545	70464152903
1546	168996745347
1547	160429275275
1548	142976782372
1549	170014476063
1550	174553795390
1551	222454633082
1552	33051947625
1553	37182176622
1554	344855543718
1555	119208146283
1556	107282165062
1557	143282458305
1558	164872223950
1559	385122746008
1560	62578713270
1561	176158811742
1562	118027566675
1563	364921287610
1564	299852086093
1565	136243056022
1566	399858131992
1567	223700692605
1568	359606250350
1569	558397733113
1570	206516426483
1571	281055726357
1572	215157908388
1573	248533440965
1574	605483702178
1575	103750795712
1576	278322616602
1577	185624758875
1578	354907107055
1579	275064432933
1580	184298726117
1581	513934851092
1582	383060493443
1583	129919858890
1584	550087135293
1585	96643867775
1586	244777442157
1587	144666194755
1588	317882759267
1589	504467037198
1590	116320889387
1591	552495066182
1592	385287103058
1593	307663309660
1594	268309573438
1595	177702480225
1596	255548544727
1597	785707760883
1598	276185251937
1599	310000481863
1600	243015237435
1601	476094473867
1602	401584755318
1603	121584681445
1604	333819870513
1605	185545828858
1606	809203672447
1607	220818051338
1608	562941500147
1609	566988071438
1610	109354338662
1611	538043378982
1612	383806176040
1613	128518326722
1614	425743230153
1615	210586309103
1616	601754613962
1617	211557516208
1618	151521188780
1619	261923049103
1620	111423209912
1621	466752047347
1622	395629846073
1623	284209749890
1624	201089339318
1625	297772201955
1626	354443118657
1627	538605605153
1628	563886417640
1629	450374566348
1630	280475819957
1631	117959421867
1632	256713987023
1633	103293294022
1634	432495163513
1635	101323871015


1637	314387243880
1638	223187612730
1639	430370370398
1640	251399472527
1641	272511992827
1642	212024898078
1643	158850706422
1645	464373757405
1646	197557993677
1647	384183995148
1648	555439323975
1649	158176828393
1650	409316753062
1651	312158180092
1652	470531332488
1653	841664786197
1654	701771647838
1655	405384059735
1656	514927351307
1657	229220533173
1658	166226961612
1659	165136905488
1660	133864375453
1661	633288781027
1662	256117920503
1663	492030454557
1664	131415761768
1665	291876048707
1666	802227020182
1667	375860209433
1668	285028014162
1669	859713724753
1670	295839732305
1671	579035340417
1672	161263436148
1673	498618428432
1674	636283516158
1675	616638723823
1676	859939543782
1677	508171256813
1678	487863089385
1679	600200255728
1680	448760176352
1681	569028734977
1682	280877509220
1684	435854938013
1685	256321846760
1687	156290145345
1688	611197905670
1689	984424116398
1690	652938228583
1691	463542106657
1692	254281560703
1693	431955803822
1694	334135265893
1695	282763122140
1696	450016651352
1697	822643098295
1698	565141146050
1699	372837607753
1700	58917239540
1701	433407295157
1702	298348323303
1703	466812660410
1705	517453687215
1706	772867305887
1707	524920422073
1708	459952950865
1709	515336777288
1710	89470349442
1712	567801655963
1713	216733386980
1715	555007406300
1716	485816451332
1717	504463536573
1718	661561148110
1719	325264034063
1720	453489992522
1722	610655512490
1723	80799633875
1724	634233966553
1725	762318092370
1726	261705340277
1727	225881516975
1728	538335755667
1729	806506374413
1730	217895151387
1731	707703825092
1732	735596463515
1733	768450016780
1734	745631868578
1735	356177753412
1736	741222736422
1737	761250576468
1738	671721527412
1740	384364766532
1741	563317414062
1742	724809604908
1743	370979739580
1744	359521940948
1745	300186232845
1746	447197130142
1747	907203956603
1748	528077281142
1749	801659775458
1750	280488680647
1751	739849772267
1752	496386470473
1753	714960674750
1754	338841041033
1755	532001966893
1756	671605907382
1757	294509651840
1759	298338286438
1761	584390565997
1762	318182662973
1763	600635385640
1764	240121693893
1766	473256996777
1767	828485975438
1769	782711361523
1770	855953593323
1771	988638208437
1772	655550748640
1773	411932055292
1774	447362234533
1775	449770326922
1776	805225566947
1777	550358556340
1778	106405396032
1780	982232544190
1781	841171099652
1785	130744752537
1786	825384984377
1787	776268942683
1788	854615602792
1789	646099785688
1790	557950608758
1791	369489065347
1792	689680764418
1794	621462178578
1795	750466894258
1797	582262097498
1798	608219715197
1799	244370452833
1800	831467803302
1802	800163087448
1806	363596492507
1807	589681107438
1808	374311870175
1810	133135498198
1811	754512505717
1815	169866686215
1816	536707956407
1817	956610710023
1818	707849433580
1819	478026230448
1820	524708956392
1822	363792713305
1823	393901596172
1825	718703397508
1827	514690190788
1828	544205194130
1830	283716126022
1832	746075980130
1836	663607308612
1837	361289615670
1838	284106487357
1840	389453919375
1841	738549390442
1843	737313907580
1846	823245280032
1847	675857800098
1848	723741177537
1850	804839675663
1851	363074768532
1853	932204536545
1857	729575524785
1860	843314053692
1862	663487911713
1863	336185917595
1865	781793275960
1866	892852657037
1870	513194527407
1871	971781746092
1872	447811616018
1874	750393104603
1875	598433097470
1880	904212211453
1881	781557375732
1885	676152407723
1887	426285083663
1890	151287342070
1892	201865544225
1893	290614184492
1895	854745621275
1897	359738661993
1898	132437316545
1900	619983477157
1901	635700745862
1903	675525244735
1905	932239203395
1912	804438922933
1915	470420475572
1918	804794472115
1925	523274691888
1926	696620005307
1932	960136903043
1935	664808242868
1936	279967183457
1937	642270715125
1950	982578277387
1953	954508482785
1955	511426336883
1958	757895739535
1968	652547157105
1971	824751367527
1979	614991456963
1984	564720846228
1986	944675774037
1987	645901356965
1994	721147884418
1997	207741061965
2006	541952390642
2016	389348710292
2023	224142765215
2029	548899132323
2032	801995327063
2048	249479281077
2072	461157712565
2073	902743525100
2074	646834943323
2080	665814402872
2086	518189243667
2090	524000762887
2093	362104284385
2096	654588515312
2105	359442044602
2163	509866874615
2175	405935625685
2183	823397649847
2228	986150450957
2377	748572833888
Code:
maxgap	k
1997    207741061965
2023    224142765215
2048    249479281077
2105    359442044602
2175    405935625685
2377    748572833888
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Old 2019-04-20, 14:07   #13
Bobby Jacobs
 
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May 2018

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114587 is followed by two consecutive record gaps of 155 and 168. The next two numbers in the sequence are 114742 and 114910. This corresponds to the twin primes (687521, 687523), (688451, 688453), and (689459, 689461).
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Old 2019-04-25, 17:47   #14
robert44444uk
 
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Jun 2003
Oxford, UK

22×3×7×23 Posts
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I thought I would start at 1e15 to make sure of no overlaps. In a couple of hours I am at 1.011e15 and have found candidates for all up to k= 2138 and many others up to k=2928.

Here's the perl code I am using - it takes advantage of danaj's fast forprimes function

Code:
#!/usr/bin/env perl
use warnings;
use strict;
use Math::GMPz;
use Math::BigFloat lib=>"GMP";
use Math::Prime::Util qw/:all/;
use v5.10;

$|=1;

my $interval = 1_000_000_000;
my $overlap = 5_000;
my $numberofiterations = 5_000;
my $from = 1_001_150_000_005_000+5_000;
my @twins;
my $mingap = 2_138;

for (my $k=0; $k <= $numberofiterations; $k++)
	{ 
	my $range = $from+$k*$interval;
	my $start = $range-$overlap;
	my $end = $range+$interval+$overlap;
	my @twins;
	forprimes 
		{ 
		if (is_prime($_+2)) {push @twins,$_;}

		}$start,$end;
	say "# of twins= ",scalar(@twins), " in the range: ", $start," ",$end;

	for (my $l=1; $l <= scalar(@twins)-1; $l++)
		{ 
		my $gap = ($twins[$l]-$twins[$l-1])/6; 
		my $first = ($twins[$l-1]+1)/6;
		my $second = ($twins[$l]+1)/6;
		if ($gap >= $mingap) {say $gap, " ", $first;}
		}

	}
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Old 2019-04-26, 07:49   #15
robert44444uk
 
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Jun 2003
Oxford, UK

22·3·7·23 Posts
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Quote:
Originally Posted by robert44444uk View Post
I thought I would start at 1e15 to make sure of no overlaps. In a couple of hours I am at 1.011e15 and have found candidates for all up to k= 2138 and many others up to k=2928.
Overnight I am at 1.004891e15 and have found values for all k to 2299, with a high gap of 3455, found at 167423264641603, equivalent to the prime pairs starting at 1004539587849619

I've started two other runs at 1.1e15 and 1.2e15
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Old 2019-04-27, 14:16   #16
robert44444uk
 
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I'm also reserving 1e12 to 1e15.
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Old 2019-04-28, 08:26   #17
robert44444uk
 
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193210 Posts
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Quote:
Originally Posted by rudy235 View Post

So in a comparison to the "gap between primes" we now can establish gaps between contiguous pairs of primes. (of the form 6k +/-1

The first gap of 1 appears at the start as we can see here

Code:
k         Gap   
1                     1
2                     1
3                     2
5                     2
7                     3
10                    2
12                    5
17                    1
18                    5
23                    2
...
So this is https://oeis.org/A160273
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Old 2019-04-29, 09:37   #18
robert44444uk
 
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Jun 2003
Oxford, UK

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Here is the table of consecutive first instance gaps between twin primes as of this morning

Code:
1	1	rudy235
2	3	rudy235
3	7	rudy235
4	103	rudy235
5	12	rudy235
6	52	rudy235
7	378	ATH
8	87	rudy235
9	313	rudy235
10	77	rudy235
11	287	rudy235
12	58	rudy235
13	192	rudy235
14	298	rudy235
15	597	ATH
16	357	rudy235
17	1075	rudy235
18	220	rudy235
19	3563	ATH
20	248	rudy235
21	2042	ATH
22	800	ATH
23	147	rudy235
24	3843	ATH
25	110	rudy235
26	3257	ATH
27	2063	ATH
28	397	ATH
29	6458	ATH
30	1755	ATH
31	6227	ATH
32	1438	ATH
33	1507	ATH
34	5638	ATH
35	980	ATH
36	13372	ATH
37	2560	ATH
38	7637	ATH
39	6018	ATH
40	2438	ATH
41	6332	ATH
42	6088	ATH
43	14542	ATH
44	11833	ATH
45	2478	ATH
46	6692	ATH
47	2233	ATH
48	9105	ATH
49	6808	ATH
50	8432	ATH
51	23277	ATH
52	7968	ATH
53	6585	ATH
54	23133	ATH
55	13815	ATH
56	25347	ATH
57	13953	ATH
58	7462	ATH
59	35798	ATH
60	31972	ATH
61	25587	ATH
62	3090	ATH
63	17475	ATH
64	90423	ATH
65	9002	ATH
66	72942	ATH
67	19033	ATH
68	17850	ATH
69	32378	ATH
70	4377	ATH
71	12952	ATH
72	23693	ATH
73	48785	ATH
74	37058	ATH
75	14845	ATH
76	58077	ATH
77	35368	ATH
78	77697	ATH
79	18308	ATH
80	55143	ATH
81	33397	ATH
82	74400	ATH
83	4070	ATH
84	24168	ATH
85	20478	ATH
86	138187	ATH
87	80868	ATH
88	22890	ATH
89	56523	ATH
90	55632	ATH
91	37942	ATH
92	81448	ATH
93	44660	ATH
94	213103	ATH
95	97545	ATH
96	354662	ATH
97	27620	ATH
98	27977	ATH
99	383148	ATH
100	44905	ATH
101	125472	ATH
102	20013	ATH
103	76967	ATH
104	293123	ATH
105	10383	ATH
106	241847	ATH
107	91303	ATH
108	89567	ATH
109	91103	ATH
110	140420	ATH
111	129022	ATH
112	149863	ATH
113	130757	ATH
114	239678	ATH
115	126070	ATH
116	334862	ATH
117	100658	ATH
118	429182	ATH
119	304243	ATH
120	140795	ATH
121	614567	ATH
122	322018	ATH
123	199937	ATH
124	535353	ATH
125	73238	ATH
126	148897	ATH
127	135550	ATH
128	410312	ATH
129	217368	ATH
130	37355	ATH
131	264492	ATH
132	150433	ATH
133	187047	ATH
134	376973	ATH
135	500720	ATH
136	828307	ATH
137	156343	ATH
138	311532	ATH
139	767163	ATH
140	505115	ATH
141	569097	ATH
142	240345	ATH
143	41995	ATH
144	989058	ATH
145	116403	ATH
146	407412	ATH
147	308028	ATH
148	802070	ATH
149	989443	ATH
150	282408	ATH
151	707262	ATH
152	277258	ATH
153	234965	ATH
154	31318	ATH
155	114587	ATH
156	607952	ATH
157	292908	ATH
158	839762	ATH
159	410923	ATH
160	154445	ATH
161	518717	ATH
162	793900	ATH
163	526890	ATH
164	443443	ATH
165	437057	ATH
166	705787	ATH
167	632180	ATH
168	114742	ATH
169	1571558	ATH
170	1074442	ATH
171	2040992	ATH
172	799050	ATH
173	1042935	ATH
174	2320048	ATH
175	613727	ATH
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1000	4718578212	ATH
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1003	6835356575	ATH
1004	19374221148	ATH
1005	714897587	ATH
1006	14257463852	ATH
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1008	10764699650	ATH
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1010	4959775740	ATH
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1012	6009418280	ATH
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1014	11991672908	ATH
1015	7504060620	ATH
1016	12499155547	ATH
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1020	13688528333	ATH
1021	12937575437	ATH
1022	6003407428	ATH
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1030	5007391888	ATH
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1034	7490905648	ATH
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1037	14585762935	ATH
1038	12918899037	ATH
1039	10528753153	ATH
1040	8643398725	ATH
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1075	7762655897	ATH
1076	19372935412	ATH
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1078	16521687485	ATH
1079	4035849583	ATH
1080	24070972425	ATH
1081	21562226522	ATH
1082	14943718163	ATH
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1088	17472321867	ATH
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1090	8186246962	ATH
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1095	40213458707	ATH
1096	28895725292	ATH
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1100	14524563518	ATH
1101	12875454587	ATH
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1103	11202556685	ATH
1104	16615629933	ATH
1105	16606921495	ATH
1106	20860243742	ATH
1107	17500043025	ATH
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1109	19821577948	ATH
1110	12852163973	ATH
1111	8412207522	ATH
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1114	23078677218	ATH
1115	9254604950	ATH
1116	15134152567	ATH
1117	24189828365	ATH
1118	29646936667	ATH
1119	34226827423	ATH
1120	18463125062	ATH
1121	58663327587	ATH
1122	10133636193	ATH
1123	26227925140	ATH
1124	42383800163	ATH
1125	38347022838	ATH
1126	20497482947	ATH
1127	8185647088	ATH
1128	29451224835	ATH
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1130	16298537952	ATH
1131	37485937562	ATH
1132	7578097340	ATH
1133	25913710302	ATH
1134	14524565688	ATH
1135	20792431795	ATH
1136	28809229977	ATH
1137	13582827433	ATH
1138	37975916165	ATH
1139	61756317343	ATH
1140	15892671560	ATH
1141	10439154257	ATH
1142	25262591533	ATH
1143	28593081655	ATH
1144	48826150303	ATH
1145	41705622962	ATH
1146	25794434892	ATH
1147	16293399008	ATH
1148	8582270267	ATH
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1150	16251349042	ATH
1151	45316370542	ATH
1152	21034892753	ATH
1153	15728686990	ATH
1154	17504175188	ATH
1155	24735441112	ATH
1156	33174606362	ATH
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1158	42035688247	ATH
1159	23092251588	ATH
1160	14998583343	ATH
1161	56801711312	ATH
1162	29812173248	ATH
1163	17756220570	ATH
1164	16797436063	ATH
1165	14435182932	ATH
1166	15804331282	ATH
1167	33766997653	ATH
1168	29664641207	ATH
1169	51537427428	ATH
1170	25320667690	ATH
1171	21420242172	ATH
1172	39668112358	ATH
1173	13791132407	ATH
1174	18761657808	ATH
1175	7073619248	ATH
1176	30537616247	ATH
1177	23153721783	ATH
1178	15554271252	ATH
1179	57841170578	ATH
1180	13255405793	ATH
1181	31234475137	ATH
1182	30924553110	ATH
1183	26168680170	ATH
1184	49319201213	ATH
1185	23812067153	ATH
1186	41936084087	ATH
1187	8939732550	ATH
1188	14330487105	ATH
1189	44674906188	ATH
1190	8031358767	ATH
1191	10603108182	ATH
1192	18835412365	ATH
1193	34746153190	ATH
1194	16560031848	ATH
1195	9391637273	ATH
1196	14528709892	ATH
1197	18242511845	ATH
1198	25046740252	ATH
1199	33650418018	ATH
1200	18359485235	ATH
1201	33051842117	ATH
1202	66002847468	ATH
1203	26782423792	ATH
1204	40847394758	ATH
1205	15928146832	ATH
1206	64999749057	ATH
1207	9988457248	ATH
1208	19986121662	ATH
1209	53500919883	ATH
1210	29408935490	ATH
1211	32056558362	ATH
1212	46819768958	ATH
1213	23359512832	ATH
1214	20331890558	ATH
1215	36293533640	ATH
1216	68006592497	ATH
1217	48101859580	ATH
1218	35070286585	ATH
1219	10774224213	ATH
1220	36726742647	ATH
1221	25678337692	ATH
1222	8695593693	ATH
1223	43949266767	ATH
1224	90705803338	ATH
1225	25905710483	ATH
1226	35530520182	ATH
1227	50758082210	ATH
1228	43848563047	ATH
1229	44351452008	ATH
1230	26404449530	ATH
1231	101208274662	ATH
1232	22556325743	ATH
1233	37269319510	ATH
1234	45484451658	ATH
1235	41601982865	ATH
1236	20875795512	ATH
1237	56829323030	ATH
1238	32867985517	ATH
1239	33904468903	ATH
1240	37951687547	ATH
1241	23524711222	ATH
1242	51497620853	ATH
1243	64872205377	ATH
1244	65789117053	ATH
1245	57057845338	ATH
1246	40884738972	ATH
1247	36050767643	ATH
1248	53024192545	ATH
1249	48725636383	ATH
1250	28543693757	ATH
1251	64133691577	ATH
1252	45976631095	ATH
1253	56015688972	ATH
1254	100761544543	ATH
1255	22544518132	ATH
1256	38237309287	ATH
1257	37685943368	ATH
1258	20020326737	ATH
1259	89817312058	ATH
1260	40437430637	ATH
1261	81952221312	ATH
1262	34386364680	ATH
1263	64312175652	ATH
1264	41031871478	ATH
1265	7595970438	ATH
1266	66001896202	ATH
1267	102711625448	ATH
1268	52055075942	ATH
1269	60464331573	ATH
1270	26072298478	ATH
1271	94947179337	ATH
1272	37482239365	ATH
1273	52338755872	ATH
1274	31605073873	ATH
1275	37883577230	ATH
1276	19485934097	ATH
1277	10846539718	ATH
1278	53318021160	ATH
1279	80634404163	ATH
1280	40551655827	ATH
1281	31260613627	ATH
1282	51115333138	ATH
1283	69732506812	ATH
1284	70023017393	ATH
1285	12347431502	ATH
1286	41906763817	ATH
1287	29873248415	ATH
1288	29314359620	ATH
1289	44402436488	ATH
1290	43944263470	ATH
1291	60328695492	ATH
1292	35124127965	ATH
1293	44978931772	ATH
1294	78981727158	ATH
1295	66748779720	ATH
1296	136174101612	ATH
1297	83379921555	ATH
1298	9634521157	ATH
1299	29706714558	ATH
1300	59173429653	ATH
1301	64935700882	ATH
1302	77686997383	ATH
1303	68044441195	ATH
1304	40831407493	ATH
1305	24030385738	ATH
1306	37042953262	ATH
1307	92308689725	ATH
1308	91890048212	ATH
1309	69089723273	ATH
1310	71131302125	ATH
1311	65562906117	ATH
1312	50485273458	ATH
1313	48624178687	ATH
1314	83852500748	ATH
1315	48882377632	ATH
1316	75146429732	ATH
1317	36692181453	ATH
1318	53992589912	ATH
1319	186438133443	ATH
1320	51173468143	ATH
1321	30530362357	ATH
1322	94707449170	ATH
1323	42450011142	ATH
1324	70051577718	ATH
1325	40320006480	ATH
1326	78720103417	ATH
1327	31520502065	ATH
1328	34445495270	ATH
1329	143446677043	ATH
1330	7287610437	ATH
1331	86046387072	ATH
1332	51772611128	ATH
1333	104162957980	ATH
1334	44438479298	ATH
1335	46412332407	ATH
1336	92479857902	ATH
1337	51780274385	ATH
1338	97017192822	ATH
1339	79715880988	ATH
1340	10849288625	ATH
1341	82083852107	ATH
1342	132223116333	ATH
1343	25083224815	ATH
1344	62994611978	ATH
1345	78773730823	ATH
1346	103895222522	ATH
1347	66824446040	ATH
1348	93470563142	ATH
1349	46032970723	ATH
1350	110940120422	ATH
1351	147555423337	ATH
1352	44028349955	ATH
1353	72388630452	ATH
1354	176010608498	ATH
1355	35415187398	ATH
1356	73721641492	ATH
1357	39997802863	ATH
1358	68875590515	ATH
1359	238021889388	Thomas11
1360	120701590522	ATH
1361	103656076897	ATH
1362	129780127055	ATH
1363	45464706810	ATH
1364	236116004983	Thomas11
1365	27432631065	ATH
1366	221323766702	Thomas11
1367	66664704428	ATH
1368	79675959260	ATH
1369	105795508798	ATH
1370	22751684835	ATH
1371	150422884902	ATH
1372	73987959848	ATH
1373	91101657665	ATH
1374	74619205678	ATH
1375	106433204497	ATH
1376	109109662112	ATH
1377	140926519020	ATH
1378	158222902962	ATH
1379	59513006413	ATH
1380	83535481318	ATH
1381	81120569787	ATH
1382	208297454633	Thomas11
1383	82037154155	ATH
1384	91105669713	ATH
1385	35954269187	ATH
1386	36654125322	ATH
1387	82436707565	ATH
1388	52106714982	ATH
1389	335699670478	Thomas11
1390	134783026287	ATH
1391	91527303572	ATH
1392	160241217295	ATH
1393	78441568602	ATH
1394	107218832948	ATH
1395	93204583073	ATH
1396	370369427962	Thomas11
1397	71529209675	ATH
1398	89717150497	ATH
1399	167763335213	ATH
1400	78769799252	ATH
1401	113406055667	ATH
1402	83591984703	ATH
1403	209318000600	Thomas11
1404	115946907303	ATH
1405	60066856568	ATH
1406	157757582267	ATH
1407	207000817860	Thomas11
1408	75688827280	ATH
1409	131711929323	ATH
1410	199574686315	ATH
1411	48278160327	ATH
1412	157757709578	ATH
1413	32135929095	ATH
1414	179075794238	ATH
1415	49037845735	ATH
1416	64167955752	ATH
1417	117577979190	ATH
1418	108513597027	ATH
1419	77078300443	ATH
1420	74951308542	ATH
1421	42119304362	ATH
1422	101735004478	ATH
1423	142152878497	ATH
1424	258224787443	Thomas11
1425	88942037140	ATH
1426	114261540757	ATH
1427	228073106628	Thomas11
1428	100389566412	ATH
1429	175835907578	ATH
1430	48978765967	ATH
1431	105371512887	ATH
1432	89851341625	ATH
1433	211777215930	Thomas11
1434	197709087693	ATH
1435	110562730142	ATH
1436	34440671017	ATH
1437	141185526520	ATH
1438	167850568830	ATH
1439	94595234198	ATH
1440	34516209527	ATH
1441	195969981917	ATH
1442	136888972125	ATH
1443	134581666552	ATH
1444	159728123338	ATH
1445	38840382823	ATH
1446	227253338462	Thomas11
1447	57301290370	ATH
1448	77920988750	ATH
1449	102332614223	ATH
1450	108516370937	ATH
1451	64095115322	ATH
1452	151527991675	ATH
1453	145272707550	ATH
1454	174799142838	ATH
1455	81631078743	ATH
1456	195778814802	ATH
1457	301410421995	Thomas11
1458	264657155665	Thomas11
1459	206268027728	Thomas11
1460	64584093742	ATH
1461	472019855132	Thomas11
1462	88475830910	ATH
1463	69741598787	ATH
1464	224994765053	Thomas11
1465	135767046975	ATH
1466	275157656942	Thomas11
1467	85661945075	ATH
1468	63139736062	ATH
1469	228677692513	Thomas11
1470	106208976653	ATH
1471	156171984272	ATH
1472	138299223690	ATH
1473	184783689402	ATH
1474	365110475063	Thomas11
1475	67410836280	ATH
1476	267918092432	Thomas11
1477	185821439738	ATH
1478	96139203857	ATH
1479	214041248743	Thomas11
1480	240784621312	Thomas11
1481	228476499377	Thomas11
1482	176427443750	ATH
1483	48773094382	ATH
1484	136319467498	ATH
1485	169809420105	ATH
1486	150516739082	ATH
1487	405178602348	Thomas11
1488	209618644275	Thomas11
1489	130741136258	ATH
1490	88340661855	ATH
1491	227858144272	Thomas11
1492	121639858400	ATH
1493	66924164457	ATH
1494	406142933448	Thomas11
1495	69782047618	ATH
1496	174973906777	ATH
1497	291869230735	Thomas11
1498	308394175852	Thomas11
1499	22339570278	ATH
1500	172823544343	ATH
1501	326074608547	Thomas11
1502	128837925765	ATH
1503	181674927455	ATH
1504	289506845778	Thomas11
1505	160823433270	ATH
1506	227491883602	Thomas11
1507	176020578323	ATH
1508	279911847660	Thomas11
1509	434412984948	Thomas11
1510	163775464003	ATH
1511	393469552312	Thomas11
1512	218349176503	Thomas11
1513	342721571537	Thomas11
1514	440909475828	Thomas11
1515	155390806498	ATH
1516	85283577452	ATH
1517	196413282468	ATH
1518	184159612907	ATH
1519	235557862248	Thomas11
1520	159460255028	ATH
1521	359489269902	Thomas11
1522	219322573465	Thomas11
1523	147488057610	ATH
1524	263159931988	Thomas11
1525	98616660978	ATH
1526	56105289977	ATH
1527	194013931730	ATH
1528	263122426457	Thomas11
1529	137914593968	ATH
1530	164942161335	ATH
1531	82643395422	ATH
1532	238243169945	Thomas11
1533	247333123902	Thomas11
1534	450299367998	Thomas11
1535	328710671530	Thomas11
1536	151913191462	ATH
1537	343278293043	Thomas11
1538	119496454527	ATH
1539	225373027793	Thomas11
1540	139666887840	ATH
1541	251898401212	Thomas11
1542	181392272663	ATH
1543	180056667807	ATH
1544	146578135928	ATH
1545	70464152903	ATH
1546	168996745347	ATH
1547	160429275275	ATH
1548	142976782372	ATH
1549	170014476063	ATH
1550	174553795390	ATH
1551	222454633082	Thomas11
1552	33051947625	ATH
1553	37182176622	ATH
1554	344855543718	Thomas11
1555	119208146283	ATH
1556	107282165062	ATH
1557	143282458305	ATH
1558	164872223950	ATH
1559	385122746008	Thomas11
1560	62578713270	ATH
1561	176158811742	ATH
1562	118027566675	ATH
1563	364921287610	Thomas11
1564	299852086093	Thomas11
1565	136243056022	ATH
1566	399858131992	Thomas11
1567	223700692605	Thomas11
1568	359606250350	Thomas11
1569	558397733113	Thomas11
1570	206516426483	Thomas11
1571	281055726357	Thomas11
1572	215157908388	Thomas11
1573	248533440965	Thomas11
1574	605483702178	Thomas11
1575	103750795712	ATH
1576	278322616602	Thomas11
1577	185624758875	ATH
1578	354907107055	Thomas11
1579	275064432933	Thomas11
1580	184298726117	ATH
1581	513934851092	Thomas11
1582	383060493443	Thomas11
1583	129919858890	ATH
1584	550087135293	Thomas11
1585	96643867775	ATH
1586	244777442157	Thomas11
1587	144666194755	ATH
1588	317882759267	Thomas11
1589	504467037198	Thomas11
1590	116320889387	ATH
1591	552495066182	Thomas11
1592	385287103058	Thomas11
1593	307663309660	Thomas11
1594	268309573438	Thomas11
1595	177702480225	ATH
1596	255548544727	Thomas11
1597	785707760883	Thomas11
1598	276185251937	Thomas11
1599	310000481863	Thomas11
1600	243015237435	Thomas11
1601	476094473867	Thomas11
1602	401584755318	Thomas11
1603	121584681445	ATH
1604	333819870513	Thomas11
1605	185545828858	ATH
1606	809203672447	Thomas11
1607	220818051338	Thomas11
1608	562941500147	Thomas11
1609	566988071438	Thomas11
1610	109354338662	ATH
1611	538043378982	Thomas11
1612	383806176040	Thomas11
1613	128518326722	ATH
1614	425743230153	Thomas11
1615	210586309103	Thomas11
1616	601754613962	Thomas11
1617	211557516208	Thomas11
1618	151521188780	ATH
1619	261923049103	Thomas11
1620	111423209912	ATH
1621	466752047347	Thomas11
1622	395629846073	Thomas11
1623	284209749890	Thomas11
1624	201089339318	Thomas11
1625	297772201955	Thomas11
1626	354443118657	Thomas11
1627	538605605153	Thomas11
1628	563886417640	Thomas11
1629	450374566348	Thomas11
1630	280475819957	Thomas11
1631	117959421867	ATH
1632	256713987023	Thomas11
1633	103293294022	ATH
1634	432495163513	Thomas11
1635	101323871015	ATH
1636	1035211477332	robert44444uk
1637	314387243880	Thomas11
1638	223187612730	Thomas11
1639	430370370398	Thomas11
1640	251399472527	Thomas11
1641	272511992827	Thomas11
1642	212024898078	Thomas11
1643	158850706422	ATH
1644	1201386008933	robert44444uk
1645	464373757405	Thomas11
1646	197557993677	ATH
1647	384183995148	Thomas11
1648	555439323975	Thomas11
1649	158176828393	ATH
1650	409316753062	Thomas11
1651	312158180092	Thomas11
1652	470531332488	Thomas11
1653	841664786197	Thomas11
1654	701771647838	Thomas11
1655	405384059735	Thomas11
1656	514927351307	Thomas11
1657	229220533173	Thomas11
1658	166226961612	ATH
1659	165136905488	ATH
1660	133864375453	ATH
1661	633288781027	Thomas11
1662	256117920503	Thomas11
1663	492030454557	Thomas11
1664	131415761768	ATH
1665	291876048707	Thomas11
1666	802227020182	Thomas11
1667	375860209433	Thomas11
1668	285028014162	Thomas11
1669	859713724753	Thomas11
1670	295839732305	Thomas11
1671	579035340417	Thomas11
1672	161263436148	ATH
1673	498618428432	Thomas11
1674	636283516158	Thomas11
1675	616638723823	Thomas11
1676	859939543782	Thomas11
1677	508171256813	Thomas11
1678	487863089385	Thomas11
1679	600200255728	Thomas11
1680	448760176352	Thomas11
1681	569028734977	Thomas11
1682	280877509220	Thomas11
1683	1019626449685	robert44444uk
1684	435854938013	Thomas11
1685	256321846760	Thomas11
1686	1025620695957	robert44444uk
1687	156290145345	ATH
1688	611197905670	Thomas11
1689	984424116398	Thomas11
1690	652938228583	Thomas11
1691	463542106657	Thomas11
1692	254281560703	Thomas11
1693	431955803822	Thomas11
1694	334135265893	Thomas11
1695	282763122140	Thomas11
1696	450016651352	Thomas11
1697	822643098295	Thomas11
1698	565141146050	Thomas11
1699	372837607753	Thomas11
1700	58917239540	ATH
1701	433407295157	Thomas11
1702	298348323303	Thomas11
1703	466812660410	Thomas11
1704	1045077852373	robert44444uk
1705	517453687215	Thomas11
1706	772867305887	Thomas11
1707	524920422073	Thomas11
1708	459952950865	Thomas11
1709	515336777288	Thomas11
1710	89470349442	ATH
1711	1437737041932	robert44444uk
1712	567801655963	Thomas11
1713	216733386980	Thomas11
1714	1039648289663	robert44444uk
1715	555007406300	Thomas11
1716	485816451332	Thomas11
1717	504463536573	Thomas11
1718	661561148110	Thomas11
1719	325264034063	Thomas11
1720	453489992522	Thomas11
1721	1253001163102	robert44444uk
1722	610655512490	Thomas11
1723	80799633875	ATH
1724	634233966553	Thomas11
1725	762318092370	Thomas11
1726	261705340277	Thomas11
1727	225881516975	Thomas11
1728	538335755667	Thomas11
1729	806506374413	Thomas11
1730	217895151387	Thomas11
1731	707703825092	Thomas11
1732	735596463515	Thomas11
1733	768450016780	Thomas11
1734	745631868578	Thomas11
1735	356177753412	Thomas11
1736	741222736422	Thomas11
1737	761250576468	Thomas11
1738	671721527412	Thomas11
1739	1214363016178	robert44444uk
1740	384364766532	Thomas11
1741	563317414062	Thomas11
1742	724809604908	Thomas11
1743	370979739580	Thomas11
1744	359521940948	Thomas11
1745	300186232845	Thomas11
1746	447197130142	Thomas11
1747	907203956603	Thomas11
1748	528077281142	Thomas11
1749	801659775458	Thomas11
1750	280488680647	Thomas11
1751	739849772267	Thomas11
1752	496386470473	Thomas11
1753	714960674750	Thomas11
1754	338841041033	Thomas11
1755	532001966893	Thomas11
1756	671605907382	Thomas11
1757	294509651840	Thomas11
1758	1278737589385	robert44444uk
1759	298338286438	Thomas11
1760	1175000358333	robert44444uk
1761	584390565997	Thomas11
1762	318182662973	Thomas11
1763	600635385640	Thomas11
1764	240121693893	Thomas11
1765	1204735995605	robert44444uk
1766	473256996777	Thomas11
1767	828485975438	Thomas11
1768	1176057612252	robert44444uk
1769	782711361523	Thomas11
1770	855953593323	Thomas11
1771	988638208437	Thomas11
1772	655550748640	Thomas11
1773	411932055292	Thomas11
1774	447362234533	Thomas11
1775	449770326922	Thomas11
1776	805225566947	Thomas11
1777	550358556340	Thomas11
1778	106405396032	ATH
And non-consecutive first instance gaps found

Code:
1780	982232544190	Thomas11
1781	841171099652	Thomas11
1782	1224262909268	robert44444uk
1783	1422392477865	robert44444uk
1784	1195747448213	robert44444uk
1785	130744752537	ATH
1786	825384984377	Thomas11
1787	776268942683	Thomas11
1788	854615602792	Thomas11
1789	646099785688	Thomas11
1790	557950608758	Thomas11
1791	369489065347	Thomas11
1792	689680764418	Thomas11
1793	1037083894142	robert44444uk
1794	621462178578	Thomas11
1795	750466894258	Thomas11
1797	582262097498	Thomas11
1798	608219715197	Thomas11
1799	244370452833	Thomas11
1800	831467803302	Thomas11
1801	1052870164587	robert44444uk
1802	800163087448	Thomas11
1803	1306998722697	robert44444uk
1804	1382597845958	robert44444uk
1805	1166113583162	robert44444uk
1806	363596492507	Thomas11
1807	589681107438	Thomas11
1808	374311870175	Thomas11
1809	1519940585608	robert44444uk
1810	133135498198	ATH
1811	754512505717	Thomas11
1812	1225517231188	robert44444uk
1813	1177115829552	robert44444uk
1814	1622632461223	robert44444uk
1815	169866686215	ATH
1816	536707956407	Thomas11
1817	956610710023	Thomas11
1818	707849433580	Thomas11
1819	478026230448	Thomas11
1820	524708956392	Thomas11
1821	1545928373782	robert44444uk
1822	363792713305	Thomas11
1823	393901596172	Thomas11
1825	718703397508	Thomas11
1826	1197393987332	robert44444uk
1827	514690190788	Thomas11
1828	544205194130	Thomas11
1829	1682087625368	robert44444uk
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1831	1024442893122	robert44444uk
1832	746075980130	Thomas11
1833	1171018474300	robert44444uk
1834	1461304039818	robert44444uk
1835	1236784093577	robert44444uk
1836	663607308612	Thomas11
1837	361289615670	Thomas11
1838	284106487357	Thomas11
1839	1320327308713	robert44444uk
1840	389453919375	Thomas11
1841	738549390442	Thomas11
1842	1084227409855	robert44444uk
1843	737313907580	Thomas11
1845	1537365969212	robert44444uk
1846	823245280032	Thomas11
1847	675857800098	Thomas11
1848	723741177537	Thomas11
1850	804839675663	Thomas11
1851	363074768532	Thomas11
1853	932204536545	Thomas11
1854	1189288485208	robert44444uk
1855	1174605355802	robert44444uk
1856	1297574083922	robert44444uk
1857	729575524785	Thomas11
1858	1123294853522	robert44444uk
1860	843314053692	Thomas11
1862	663487911713	Thomas11
1863	336185917595	Thomas11
1864	1417574651223	robert44444uk
1865	781793275960	Thomas11
1866	892852657037	Thomas11
1867	1444042188138	robert44444uk
1870	513194527407	Thomas11
1871	971781746092	Thomas11
1872	447811616018	Thomas11
1873	1348365096302	robert44444uk
1874	750393104603	Thomas11
1875	598433097470	Thomas11
1876	1064530504212	robert44444uk
1877	1040113573768	robert44444uk
1878	1694010994357	robert44444uk
1880	904212211453	Thomas11
1881	781557375732	Thomas11
1882	1375789467133	robert44444uk
1883	1086233157432	robert44444uk
1885	676152407723	Thomas11
1887	426285083663	Thomas11
1888	1709285542095	robert44444uk
1890	151287342070	ATH
1891	1247181224132	robert44444uk
1892	201865544225	Thomas11
1893	290614184492	Thomas11
1894	1666811164798	robert44444uk
1895	854745621275	Thomas11
1897	359738661993	Thomas11
1898	132437316545	ATH
1900	619983477157	Thomas11
1901	635700745862	Thomas11
1902	1068189680728	robert44444uk
1903	675525244735	Thomas11
1905	932239203395	Thomas11
1906	1279486288362	robert44444uk
1907	1202177228578	robert44444uk
1908	1222826404275	robert44444uk
1910	1551507663815	robert44444uk
1911	1670504219247	robert44444uk
1912	804438922933	Thomas11
1914	1488017568723	robert44444uk
1915	470420475572	Thomas11
1917	1165423934903	robert44444uk
1918	804794472115	Thomas11
1919	1572478640448	robert44444uk
1923	1395252802692	robert44444uk
1924	1696444081623	robert44444uk
1925	523274691888	Thomas11
1926	696620005307	Thomas11
1928	1181669865490	robert44444uk
1932	960136903043	Thomas11
1933	1521234330052	robert44444uk
1934	1368396113168	robert44444uk
1935	664808242868	Thomas11
1936	279967183457	Thomas11
1937	642270715125	Thomas11
1940	1435690154763	robert44444uk
1941	1675555446682	robert44444uk
1943	1042310309512	robert44444uk
1945	1194795033583	robert44444uk
1946	1267188286017	robert44444uk
1947	1501956317263	robert44444uk
1950	982578277387	Thomas11
1953	954508482785	Thomas11
1955	511426336883	Thomas11
1958	757895739535	Thomas11
1960	1603135467927	robert44444uk
1964	1717072682918	robert44444uk
1968	652547157105	Thomas11
1969	1116171078598	robert44444uk
1971	824751367527	Thomas11
1972	1484101843865	robert44444uk
1976	1629196186112	robert44444uk
1979	614991456963	Thomas11
1984	564720846228	Thomas11
1985	1432632140313	robert44444uk
1986	944675774037	Thomas11
1987	645901356965	Thomas11
1990	1178585706107	robert44444uk
1993	1333290080990	robert44444uk
1994	721147884418	Thomas11
1995	1081507694722	robert44444uk
1997	207741061965	Thomas11
2002	1418099818398	robert44444uk
2006	541952390642	Thomas11
2007	1294793177503	robert44444uk
2012	1355880364715	robert44444uk
2013	1605983743805	robert44444uk
2015	1366762534618	robert44444uk
2016	389348710292	Thomas11
2018	1320531199235	robert44444uk
2020	1110837159180	robert44444uk
2023	224142765215	Thomas11
2027	1513315229030	robert44444uk
2028	1693568705267	robert44444uk
2029	548899132323	Thomas11
2030	1159591145647	robert44444uk
2031	1505148947332	robert44444uk
2032	801995327063	Thomas11
2040	1457630950535	robert44444uk
2042	1102539778813	robert44444uk
2045	1654904607582	robert44444uk
2048	249479281077	Thomas11
2057	1327506760473	robert44444uk
2060	1506078160848	robert44444uk
2068	1462559617500	robert44444uk
2070	1042838539733	robert44444uk
2072	461157712565	Thomas11
2073	902743525100	Thomas11
2074	646834943323	Thomas11
2075	1226421246838	robert44444uk
2080	665814402872	Thomas11
2081	1557362375067	robert44444uk
2086	518189243667	Thomas11
2090	524000762887	Thomas11
2093	362104284385	Thomas11
2096	654588515312	Thomas11
2103	1654754653500	robert44444uk
2105	359442044602	Thomas11
2109	1543363000013	robert44444uk
2110	1612044398132	robert44444uk
2113	1096035536635	robert44444uk
2125	1542932450388	robert44444uk
2134	1576063926063	robert44444uk
2163	509866874615	Thomas11
2175	405935625685	Thomas11
2183	823397649847	Thomas11
2198	1225499177052	robert44444uk
2219	1346519039873	robert44444uk
2228	986150450957	Thomas11
2255	1108236319297	robert44444uk
2267	1360918509703	robert44444uk
2297	1379963970150	robert44444uk
2299	1646969192138	robert44444uk
2305	1094577877207	robert44444uk
2357	1606989456770	robert44444uk
2377	748572833888	Thomas11
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Old 2019-04-29, 13:00   #19
robert44444uk
 
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Here's an interesting graph which shows the frequency of all gaps between consecutive twin prime pairs. The population of gaps represents all k between 1 and approx. 4.43e7.
Not all gap lengths are equal in frequency. Those that do relatively poorly are lengths which are highly composite, especially with prime factors 2 and 3. Those that do well are multiples of 5,7,11,13...

In the graph -
  • red bars are multiples of 5, and those that perform really well are also multiples of 7,11,13..
  • green bars are multiples of 7
  • yellow bars are multiples of 11
  • black bars represent primes >11
  • blue bars are all others
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Old 2019-04-29, 14:07   #20
robert44444uk
 
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22·3·7·23 Posts
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Regarding the graph above, I not at all sure about gaps 0mod11, 0mod13 etc.. Here are some stats on the first 1.666e9k

There were 27,412,678 twin prime gaps, of which

8,632,028 were 0 mod5
3,221,541 were 1 mod5
6,415,438 were 2 mod5
6,086,566 were 3 mod5
3,057,104 were 4 mod5

So the 0mod5 case is made.


4,955,717 were 0 mod7
3,481,004 were 1 mod7
4,418,659 were 2 mod7
3,464,620 were 3 mod7
3,288,570 were 4 mod7
4,510,772 were 5 mod7
3,293,335 were 6 mod7

The 0mod 7 was not at all dominant in the early stages but appears to be in ascendancy the higher the k.

2,526,237 were 0 mod11
2,459,124 were 1 mod11
2,766,048 were 2 mod11
2,398,069 were 3 mod11
2,707,618 were 4 mod11
2,537,710 were 5 mod11
2,303,639 were 6 mod11
2,833,137 were 7 mod11
2,370,441 were 8 mod11
2,171,210 were 9 mod11
2,339,444 were 10 mod11

0mod13 does not show positive tendencies either.

For the small range from 1e9 covering an additional 1.667e5 k continues to show strong bias towards 0mod5, and a lesser bias towards 0mod7 but none towards 0mod11.

There is some proper modular maths going on here to show such strong bias towards 0mod5, and a strong negative bias towards 1mod5 and 4mod5

Last fiddled with by robert44444uk on 2019-04-29 at 14:12
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Old 2019-04-29, 19:00   #21
mart_r
 
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Dec 2008
you know...around...

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Quote:
Originally Posted by robert44444uk View Post
There is some proper modular maths going on here to show such strong bias towards 0mod5, and a strong negative bias towards 1mod5 and 4mod5
For a first impression, look at p+{0,2,6*k,6*k+2} for k=1..5:
Code:
                 number of open residues mod...
k%5: pattern   : 5 7 11 13
  0: 0 2 30 32 : 3 4  7  9 = 756 in 30030
  1: 0 2  6  8 : 1 3  7  9 = 189 in 30030
  2: 0 2 12 14 : 2 4  7  9 = 504 in 30030
  3: 0 2 18 20 : 2 3  7  9 = 378 in 30030
  4: 0 2 24 26 : 1 3  8 10 = 240 in 30030
Here the value for 0 mod 5 would be overestimated (27412678/2067*756=10026117 which is 16% more than the actual count), but if you look further, you see why this is so, for example p+{0,2,60,62} is also k=0 mod 5, but here there are only 3*3*7*9 = 567 in 30030 possibilities.

Continuing this line of argumentation should yield good approximations. Maybe I can work out some details tomorrow.
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Old 2019-04-30, 07:43   #22
robert44444uk
 
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Jun 2003
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Quote:
Originally Posted by mart_r View Post
For a first impression, look at p+{0,2,6*k,6*k+2} for k=1..5:
Code:
                 number of open residues mod...
k%5: pattern   : 5 7 11 13
  0: 0 2 30 32 : 3 4  7  9 = 756 in 30030
  1: 0 2  6  8 : 1 3  7  9 = 189 in 30030
  2: 0 2 12 14 : 2 4  7  9 = 504 in 30030
  3: 0 2 18 20 : 2 3  7  9 = 378 in 30030
  4: 0 2 24 26 : 1 3  8 10 = 240 in 30030
Here the value for 0 mod 5 would be overestimated (27412678/2067*756=10026117 which is 16% more than the actual count), but if you look further, you see why this is so, for example p+{0,2,60,62} is also k=0 mod 5, but here there are only 3*3*7*9 = 567 in 30030 possibilities.

Continuing this line of argumentation should yield good approximations. Maybe I can work out some details tomorrow.
Beautifully explained, and there is the pattern we are seeing! So 0mod7 is not likely to be so dominant. One could imagine a race between mod11's

Last fiddled with by robert44444uk on 2019-04-30 at 07:56
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