The best known packings of equal circles in a 1x0.70000 rectangle (complete up to N = 437)

Last update: 23-Aug-2010


Overview    Download    Results    History of updates    References

Overview

1-6   7-12   13-18   19-24   25-30   31-36   37-42   43-48   49-54   55-60   61-66   67-72   73-78   79-84   85-90   91-96   97-102   103-108   109-114   115-120   121-126   127-132   133-138   139-144   145-150   151-156   157-162   163-168   169-174   175-180   181-186   187-192   193-198   199-204   205-210   211-216   217-222   223-228   229-234   235-240   241-246   247-252   253-258   259-264   265-270   271-276   277-282   283-288   289-294   295-300   301-306   307-312   313-318   319-324   325-330   331-336   337-342   343-348   349-354   355-360   361-366   367-372   373-378   379-384   385-390   391-396   397-402   403-408   409-414   415-420   421-426   427-432   433-437  


Download

You may download ASCII files which contain all the values of radius, ratio etc. by using the links given in the table header below.
All coordinates of all packings are packed as ASCII files here.
All packings are stored as nice PDF files here.
All contact graphs of all packings are stored as nice PDF files here.
  For industrial applications, for instance if a machine has to do an important job at every circle center,
it is useful to know a tour visiting each of the circle centers once which is of minimal length.
This problem is known as the "Traveling Salesman Problem" (TSP). Thus (very near) optimal tours are provided for every packing.
All optimal TSP tours of all packings are stored as nice PDF files here.


Results

The table below summarizes the current status of the search.
Please use the links in the following table to view a picture for a certain configuration.
Furthermore, note that for certain values of N several distinct optimal configurations exist; however, only one is shown here.
Proven optimal packings are indicated by a radius in bold face type.

Legend:
Please note that all packings (including their coordinates, of course) are normalized such that their width (i.e. the horizontal dimension) is equal 1.
N
the number of circles; colors correspond to active researchers in the past, see "References" at the bottom of the page
radius
of the circles in the rectangle
ratio
= 0.7/radius
density
ratio of total area occupied by the circles to container area (for an infinite hexagonal packing you get the well-known value ρ = Pi/(2*sqrt(3))=0.90689968211)
contacts
number of contacts between circles and container and between the circles themselves, respectively
loose
number of circles that have still degrees of freedom for a movement inside the container (so called "rattlers")
boundary
number of circles that have contact to the container (rattlers too if possible)
symmetry group
of the packing (Schönfliess notation); if field is empty then the packing has symmetry element C1
reference
for the best known packing so far
records
the sequence of N 's that establish density records

N radius ratio density contacts loose boundary symmetry group reference
1 0.350000000000000000000000000000 2.0000000000000000000000000000 0.549778714378213816730962592079 2 1 D2
2 0.258392021690038395743267170844 2.7090619726630150952475516253 0.599294135962252992935427609402 5 2 C2 [1]
3 0.205892565390258373770817432927 3.3998313570632691058924699959 0.570761166886979781001533940206 7 3 D1 [1]
4 0.190231221598761233193745028355 3.6797324546254100826353999853 0.649643978878804452034307017422 9 4 C2 [1]
5 0.175717864185549522550153430972 3.9836587090589379605214953272 0.692873049057230509982683115283 12 4 D1 [1]
6 0.167158248695502731670384586987 4.1876485633390880294587841330 0.752417188041198686214118502096 14 5 D1 [1]
7 0.146332224376070850817050427221 4.7836353406411303326229721767 0.672713001395278897395890240417 15 6 D1 [2]
8 0.139271215442889792000684604206 5.0261642204669728330875789798 0.696409282466705245641168056304 19 6 C2 [1]
9 0.131686637214672965358289152472 5.3156494448170452623524851392 0.700450996082229152369351193589 19 7 D1 [1]
10 0.127324919471876015312873323879 5.4977454759326866064019057811 0.727576482728854635397122204654 23 6 D1 [1]
11 0.125147988256670698128060981604 5.5933779659673295952712543392 0.773200744169224442075527509566 27 6 D1 [1]
12 0.118556819856559100849105133863 5.9043419083518275222656666228 0.756983061094382624796410623485 28 9 D1 [1]
13 0.113363312053730327942160413968 6.1748372318923071914215447696 0.749791023549142033313927757984 30 8 C2 [1]
14 0.108161900810807857786497088150 6.4717797556499111078436676819 0.735069647208638082906966214993 27 1 9 [1]
15 0.104896986540439113264399052252 6.6732136268770805756673147428 0.740745660318468344378122074874 33 9 D1 [1]
16 0.100846457631309104197582078435 6.9412452994548796888864676920 0.730286227942310903408887512992 32 1 11 D1 [1]
17 0.098636659915971426480019342099 7.0967528766315690418468145929 0.742296596261561830623912088731 33 1 11 [1]
18 0.097913275434164037349274546765 7.1491837740702832062120483564 0.774475164892785176234233209876 37 11 D1 [1]
19 0.095098067879851095997843276275 7.3608225235910655804685320659 0.771167681169309913341089024410 39 12 [1]
20 0.092857142857142857142857142857 7.5384615384615384615384615385 0.773949210578243523761205148360 45 12 C2 [1]
21 0.089253147700753232893098243002 7.8428606501022316256343878968 0.750789534374644117692860879620 41 2 12 [1]
22 0.088021174862587805588184691571 7.9526318649210101536150044143 0.764977806356775311860109209139 54 12 [1]
23 0.087262813098028571559713327287 8.0217446028658259637381070184 0.786028127426960508041291588761 45 1 12 [2]
24 0.084200604058685361468391373641 8.3134795507181925619681178740 0.763648474816743365953555331118 52 12 C2 [1]
25 0.082757390797390305651343328240 8.4584590361695404339044919288 0.768431962156427395037137637717 60 12 C2 [1]
26 0.080919062810375005253994914123 8.6506192198539622748041238593 0.764058953098059923153346454612 56 16 D2 [1]
27 0.080024385703091483748519100802 8.7473336264917777463279673629 0.775997451352477481594040599893 54 1 15 D1 [1]
28 0.079604774283339131262714745062 8.7934424323404732034281893644 0.796320863881992335107101722910 58 14 D2 [1]
29 0.077936804406052390081579370517 8.9816358950642276525306959682 0.790560333972708863441323796608 66 2 15 [1]
30 0.077272727272727272727272727273 9.0588235294117647058823529412 0.803943574575664326371638669807 66 15 D1 [1]
31 0.074848562871263167211984873834 9.3522169718071246404125484441 0.779435996263354436455591105678 58 3 15 [2]
32 0.074023210305284956740704566116 9.4564934040698156729462050824 0.786932792115545119943176458036 62 2 17 [1]
33 0.072729408259063972041912369870 9.6247173840131145398119909395 0.783404173215752149269807218027 85 15 D1 [1]
34 0.071756184271447685637716992903 9.7552567365059175397861277576 0.785686738253593303510226843662 79 16 [1]
35 0.070334536151109295087018593032 9.9524364317423569326034909330 0.777064613810856598605363053905 77 17 D1 [1]
36 0.070041043140013981819522063390 9.9941401301045823307026016204 0.792609994827110807458222317773 90 1 15 [2]
37 0.069602842206588458016803833737 10.0570605712095400416584247261 0.804465650702441642643807253451 75 16 [2]
38 0.067738478320275643202916882003 10.3338607148852106297809746926 0.782539560543162001755029678103 77 1 16 [2]
39 0.067133324617144897372505466462 10.4270122773158465714293747624 0.788846948573877976437936726977 88 1 17 [1]
40 0.066334648857052592918875412100 10.5525545406663764764988235305 0.789937430693876735849456881038 88 2 18 [1]
41 0.066031090064732256684994375895 10.6010668506875930639113450733 0.802292298075621692883043158604 86 1 18 [1]
42 0.065904688508696891142503952863 10.6213991119558489096300307501 0.818716883632163775655684029308 85 18 C2 [1]
43 0.064260797556073890506673162019 10.8931109886891909043488617797 0.796915935052990737144507114488 91 2 20 [1]
44 0.063668014876580696071888410239 10.9945315769140507087958562853 0.800473810261803910445090237371 90 2 16 [2]
45 0.063216340383991832998506773965 11.0730864163921108058253094983 0.807092010132168873528309780220 91 17 D1 [1]
46 0.062625942915089805909080484752 11.1774764165879576772066975811 0.809688964061081702802222305935 105 1 18 [2]
47 0.061433993094167758077339192797 11.3943431762122355994712168185 0.796099185782610558461608778682 93 1 20 [2]
48 0.060545277328947515494466862204 11.5615954023439667679892050996 0.789684503621840816416164644379 91 4 21 D1 [2]
49 0.059885888727617844595021122675 11.6888972489637254442372482502 0.788672886722133105021714864095 106 2 18 [1]
50 0.059391831265027711718710105853 11.7861326227902999146237427113 0.791544379980354960401453055630 100 1 20 [2]
51 0.058844562023254324380899729144 11.8957466235091094613624022687 0.792564614054717064145310152102 109 20 D1 [1]
52 0.058432840218023793774310242289 11.9795648711952015171017863227 0.796836407967860320131264072576 125 1 20 [1]
53 0.058051238682323110954158195238 12.0583128954516772289467568028 0.801587035229703985271119446040 122 2 20 D1 [1]
54 0.057490648556709649878292975831 12.1758932552223577116349828041 0.801013818861617428687205756765 103 1 21 [1]
55 0.057246621739955734089123546245 12.2277957846974477994484516831 0.808936158300755664776280389358 111 19 D1 [1]
56 0.057079044963796210194047663863 12.2636950293052772183779657060 0.818829075788844380379240737283 136 21 D1 [1]
57 0.056574708487379293427803106219 12.3730200069198103763849973171 0.818787752902802549005130228887 109 3 24 [1]
58 0.056073726692250897353553684481 12.4835647867994553129366322951 0.818462276518902051508045872433 100 4 21 [1]
59 0.055737024349492942255091535159 12.5589768770346658539123599987 0.822605109371721661553633237491 134 4 19 [1]
60 0.055614095090467682713072238960 12.5867372086394114983978285600 0.832861589192472659660409375893 160 21 D1 [2]
61 0.054524912520642854945880597708 12.8381682361247907611197757332 0.813901076008580054095964965217 124 2 22 [2]
62 0.053792657419078793503436786729 13.0129284104066036712074852404 0.805173589385276563788898172054 123 5 23 [1]
63 0.053333972215302442504605623570 13.1248427770237943766469541185 0.804266987660640843643670254579 135 23 D1 [1]
64 0.052916789201406106279826141222 13.2283158249782770653623580506 0.804301310763880843428101132057 128 3 24 [1]
65 0.052477091876488782448099789269 13.3391538091999291440606282675 0.803349834940049488197733390975 134 1 23 [1]
66 0.052312687566895883272234365838 13.3810750805884550158496569941 0.810606035864427530379255672360 133 21 D1 [1]
67 0.051763086763323929617498142711 13.5231502557141938175858894423 0.805688135668169482885722669090 143 4 22 [1]
68 0.051381774483884842761931025943 13.6235076937552043598437328693 0.805710350339960104777919158725 148 5 23 [1]
69 0.051070764921551307901946410908 13.7064718156317961404170921689 0.807691753310035147990474310122 139 2 23 [1]
70 0.050726199215756024399565471827 13.7995751864368650876342911340 0.808378061305593140577340924603 165 3 25 [1]
71 0.050589655296940238398790261317 13.8368209052086064181697136690 0.815518132866510836276102383542 125 8 24 [1]
72 0.050353753263639442842334740131 13.9016449545474409740850658913 0.819309563240016142195174650360 175 1 24 [1]
73 0.050007651343488671782457857521 13.9978579516141312936403630402 0.819308779655700315639096734128 115 4 20 [1]
74 0.049784498252947302710255735743 14.0606016845526639519218529029 0.823136427141637724456128529669 179 1 25 [1]
75 0.049700465673074189985757373681 14.0843750761722381098402404259 0.831445930299074020375182228788 151 2 25 [1]
76 0.049666378877808812673420637250 14.0940413981491916829655005294 0.841376580498764293299806526802 153 25 D1 [1]
77 0.048974373533981839717857197960 14.2931894680447750176924237553 0.828858387334917572532173057192 121 7 24 [1]
78 0.048446874110363620837894029312 14.4488166234497664516668238868 0.821633156912276860913703073797 88 17 25 [1]
79 0.048149737844292085990957543317 14.5379815413259108430002555747 0.821990461197023137881246681043 194 2 27 [1]
80 0.047917341844487866755766392500 14.6084898088002755587565184493 0.824379637257284597954225762476 170 26 C2 [1]
81 0.047361148031253469630782707348 14.7800471293067525707500644639 0.815419877008945660157865406517 175 5 24 [1]
82 0.047106955181550783773475602849 14.8598014306420655622661013571 0.816649598705023311904065036672 161 4 26 [1]
83 0.046926542843980016372992817587 14.9169309643657178161522734303 0.820289297883231303512305891383 188 1 24 [1]
84 0.046821660771319938391478951181 14.9503453843477680888193160267 0.826465534074711689798250800396 175 24 C2 [1]
85 0.046228763708287895283345802490 15.1420878225757952024848580228 0.815258465208940757290035234685 158 8 29 [1]
86 0.045906287015557137508610833570 15.2484560505356852848457876395 0.813382115964910524937948294243 159 7 28 [1]
87 0.045690830700752213605133206804 15.3203605463989915237293479226 0.815134347126604913524220342173 126 18 26 [1]
88 0.045421536735251749168288160429 15.4111914812589012101433799822 0.814813379875681247814322985092 176 4 27 [1]
89 0.045284248342325280158819939359 15.4579136371739105280846832946 0.819098567094976958038506359658 180 5 27 [1]
90 0.045063618475774893497000040657 15.5335950302415904302508109162 0.820250426413487275639553770211 182 4 25 [1]
91 0.044827999845571342380733485239 15.6152405284964895426447463310 0.820714201660865614019130596629 206 4 28 [1]
92 0.044694261550766240052515847562 15.6619658925318830876176531565 0.824789630703151991091026874474 168 5 29 [1]
93 0.044582386076288906664750984499 15.7012681825097341606232238070 0.829585969122518331055637695450 189 3 28 [1]
94 0.044541654888297572167936301153 15.7156262324665216204422299516 0.836974802553667869358112830025 162 5 28 [1]
95 0.044471547066723449707905736020 15.7404013615659945755721155754 0.843218087213100157159164918177 215 28 D1 [1]
96 0.044172292072017878966619659190 15.8470382034676833315162033585 0.840664940820900370047671493801 183 4 30 [1]
97 0.043822862131955106163245636846 15.9733975816601988780262078789 0.836036125679141758835896373983 245 2 29 [1]
98 0.043672216580868364443683718263 16.0284971728833971396599777967 0.838857860663190548770647327403 236 2 29 [1]
99 0.043567405950144131719933107167 16.0670571206611906950805186971 0.843355015337566028157369237329 209 29 D1 [1]
100 0.042967348623505254471836570977 16.2914404175515160196011762983 0.828569502267025531545393711748 190 10 33 [1]
101 0.042742738777919416281004367869 16.3770506994655859310490849010 0.828128821051003061370312251300 199 4 32 [1]
102 0.042436365898685296402132709476 16.4952861814608496457190076952 0.824381761937775220724054212515 201 8 34 [1]
103 0.042199237268620921335294431688 16.5879775395967986393767612161 0.823186539033720930669692809280 210 5 29 [1]
104 0.042169849119418017960878665350 16.5995376938085813287708100516 0.830021355208018233968368283975 82 46 28 [1]
105 0.041798275534226474010642437486 16.7471023876763943377191716571 0.823299517294767726889008022163 245 7 30 [1]
106 0.041529313910682627607061188446 16.8555637953830556508871172276 0.820478510352359040317867675869 260 4 30 [1]
107 0.041364669953437850897298662365 16.9226540617380760353239299095 0.821664904120114849349577405390 171 8 29 [1]
108 0.041196411726600244444492150009 16.9917711436992638374067815032 0.822610726140471166923663047229 273 2 30 [1]
109 0.041012318528019390539436847718 17.0680426058274137435944842187 0.822824044391879147783462283393 210 4 30 [1]
110 0.040947907249312236227783657150 17.0948907287992659632993861871 0.827766677767389225374527216662 206 2 28 [1]
111 0.040906106126673551107392170901 17.1123596519384327381858649517 0.833587306895814088782506742340 174 8 29 [1]
112 0.040888897391999618748627921058 17.1195616572669680282087275676 0.840389571296255473953234135499 225 28 C2 [1]
113 0.040441912936944149392680566248 17.3087757023615469150460366833 0.829456578581757121073972338453 233 1 31 [1]
114 0.040426669910353026609636172381 17.3153020407632986168393268973 0.836166224224912303521240094581 64 63 29 [1]
115 0.040284969930811727765555521282 17.3762075831812653867525269056 0.837598248787645979280754208151 292 1 31 [1]
116 0.040045076786409452710128546772 17.4803011050179046930667405725 0.834849292440797012162498709297 273 3 32 [1]
117 0.039964438557230056806858624622 17.5155719752595253115160479099 0.838658449188428383854609878674 205 6 32 [1]
118 0.039891003952282283345203008251 17.5478160649288675435887859010 0.842720915848378728354833297618 118 42 29 [1]
119 0.039859129485117269265294164158 17.5618486665989096653163058758 0.848505014409075943588617497828 185 10 32 [1]
120 0.039847829197763949498991351103 17.5668289613949739751509594130 0.855150222576688550792774235381 241 32 C2 [1]
121 0.039286482743469243760163776373 17.8178332881266630041294491351 0.838153379973296871852609507072 239 5 34 [1]
122 0.039127305347762422272966200125 17.8903196572935217680323837576 0.838246101890763624954882279228 209 14 28 [1]
123 0.038919064374967975673584884793 17.9860438898481611029336335617 0.836145249020652641371522269933 233 13 37 [1]
124 0.038725198679952218947591542419 18.0760854394889238663997469581 0.834566267737372736915817610680 253 6 32 [1]
125 0.038637048724904784623213337153 18.1173258077755794740286308609 0.837470922298430391702585862607 264 5 33 [1]
126 0.038528271489763310787874822984 18.1684766259494680076059304686 0.839424090534609373939487930717 118 41 31 [1]
127 0.038350058854159177920148387632 18.2529054951915130760182467678 0.838277140342018681857234911417 76 68 31 [1]
128 0.038302570481042099629017838923 18.2755358506935138435098070200 0.842786641253637924346592145426 258 30 D2 [1]
129 0.037874383606925062659619967876 18.4821489707890575489882052406 0.830486716008999471496871109686 293 6 32 [1]
130 0.037686828043399282983284448792 18.5741288493129781551442458198 0.828656148573081177703522120493 269 5 33 [1]
131 0.037542552000919332387449117158 18.6455092339715898770208359482 0.828649190262698551537518850832 277 4 34 [1]
132 0.037394638467319262866769724869 18.7192610676463485609056450330 0.828408299931354073387249752164 288 3 32 [1]
133 0.037218331466544864304962072641 18.8079361007685703050609630011 0.826831992604906637697821395369 324 1 34 [1]
134 0.037126896601780637809023846371 18.8542556494319915640247706798 0.828960674077115992390596299642 297 34 [1]
135 0.037058276433791298394105125726 18.8891677477398947842003401641 0.832062662832715185005321743476 289 2 35 [1]
136 0.036965681182585385750308235271 18.9364831813182317291756798471 0.834042474537730245355058165423 240 10 36 [1]
137 0.036884352548964042679140134613 18.9782374265821467970727458798 0.836482248072077442919945188441 327 4 35 [1]
138 0.036825022506139029059837700496 19.0088139086216264775377875620 0.839879461217244412628394224029 353 2 34 D1 [1]
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290 0.025576748031841497029253119597 27.3686083597704656673935491415 0.851413399335334679698770664355 446 30 41 [1]
291 0.025533446519243149337257656583 27.4150220759445316600085288536 0.851458924417053464985827316840 759 4 51 [1]
292 0.025511021209684828718805291137 27.4391210859977965949694209358 0.852884794578011821184678670504 326 67 47 [1]
293 0.025506062271840072768117720898 27.4444558528673349839402637116 0.855472955173404389311656263609 75 229 55 [1]
294 0.025433967184201990678261909940 27.5222498688602838169681877409 0.853546874345249889322842238518 743 7 50 [1]
295 0.025383463717889775212755253456 27.5770087084944799383245592217 0.853052217323901068625543066817 740 4 51 [1]
296 0.025374385974072111747285919546 27.5868744455637034080836244451 0.855331816511078843472988038393 587 35 54 [1]
297 0.025337676739791747600003977021 27.6268423182098484778868834885 0.855740061850341693008612431156 741 4 53 [1]
298 0.025304544132583077148733316528 27.6630156359408122024820232926 0.856377271252050067511015342390 752 8 53 [1]
299 0.025249989843258255490793034400 27.7227834286396722238668688435 0.855550080564480339477783021694 788 6 52 [1]
300 0.025230936528290507526890918540 27.7437184789045037447821039619 0.857116448547502494620386704552 542 7 56 [1]
301 0.025225402945813599535726987521 27.7498044928622951655328875560 0.859596330498006040674911953541 476 14 49 [1]
302 0.025225232885989289930131008788 27.7499915724780914258000146210 0.862440503668120873189364260622 141 188 52 [1]
303 0.025186854961595985225180496362 27.7922750207334318048876583381 0.862665328835858392487877918333 553 15 50 [1]
304 0.025180330607564362651343222834 27.7994761430858529290451185436 0.865064065895855049896893143862 823 1 52 [1]
305 0.025085584383294077341672624093 27.9044725171389652653555635389 0.861390578379631781042691018901 558 20 51 [1]
306 0.025065750889214232977706212887 27.9265521744736274982212682408 0.862848796286681671138787296699 567 9 55 [1]
307 0.025030664648400564308261727690 27.9656976685486989253974355587 0.863246789102383693042483894456 803 3 52 [1]
308 0.025020480591271075251814122990 27.9770805139614236858546484831 0.865354076401359165990313482558 444 21 50 [1]
309 0.025009592782727846807908299189 27.9892602043258693113321010842 0.867408258954059512984330834948 275 113 52 [1]
310 0.025000108518022504206407040761 27.9998784603423650209195624245 0.869555515565052185261776981305 147 209 51 [1]
311 0.025000012962587972396114175869 27.9999854819089985976240002548 0.872353864695702206611513431777 152 187 62 [1]
312 0.025000118092511828942649878031 27.9998677370115223254853799257 0.875166221487619604565800263210 864 1 52 [2]
313 0.024835197893705631182021457816 28.1858031893279916527094094650 0.866425887892876713742361570382 767 1 54 [2]
314 0.024709312520450022739819050351 28.3294000762126879991508866277 0.860404762082122046562713368679 640 17 59 [1]
315 0.024654723094324338061973694548 28.3921258138625822904250747305 0.859335285389397642505380065161 596 22 59 [1]
316 0.024609832600810932915224842718 28.4439155419908826461958185502 0.858926959765580826682936374937 647 12 57 [1]
317 0.024551334189886278183393711321 28.5116887980922555738561360353 0.857553632137832431897063015249 622 16 55 [1]
318 0.024523853475853010505161126123 28.5436381639305964247794121760 0.858334122547469858844423629581 570 20 58 [1]
319 0.024497468496362584697151646386 28.5743810673309730335495666848 0.859181528173827522945908790145 650 9 56 [1]
320 0.024451813301075792629333497396 28.6277337136875037024879014786 0.858665380103136240861053307867 460 46 45 [1]
321 0.024325622010223022611787834421 28.7762425851153907336911407267 0.852481127487860130544155899794 665 9 59 [1]
322 0.024290253519281921936939618379 28.8181430236753525791705218280 0.852651969526151968695214794852 611 14 55 [1]
323 0.024233083758085976229144960436 28.8861296807273833669493125370 0.851278611435589678094068963112 726 11 58 [1]
324 0.024202560065421772323949376446 28.9225601799080284434511327187 0.851764344312172009585552783954 591 20 57 [1]
325 0.024166123069577285927218675426 28.9661687968985588380109292509 0.851822602114691670717640954617 615 8 56 [1]
326 0.024149304630295529615893474700 28.9863418726286396396427053554 0.853254706711303398539292375427 636 7 55 [1]
327 0.024138065520629440702546149448 28.9998384254011388286177245742 0.855075590460310577668540435873 523 11 41 [1]
328 0.024133787915369222796671788408 29.0049785162078109989407608653 0.857386537779339213351576417499 131 223 52 [1]
329 0.024038319878633964296189349988 29.1201716065931307556798913126 0.853210027097073874429365822618 266 133 47 [1]
330 0.024012829716813352981041353937 29.1510833273378250060100026063 0.853989350324903541444292224383 598 22 54 [1]
331 0.024001960176109461739042139625 29.1642847027448391296138229131 0.855801903502494238949159171123 560 39 56 [1]
332 0.023982382278513584575501119968 29.1880928204179068477418247351 0.856987641496279391209810420981 543 15 50 [1]
333 0.023978895272694527991521186556 29.1923373466295818173268692260 0.859318987434614673155584595422 392 85 57 [1]
334 0.023976070860800200029782719409 29.1957762414052833251468744019 0.861696495072214503800152743570 262 153 58 [1]
335 0.023974631709267674259634570598 29.1975288083114458779474461947 0.864172672301185082049728322859 703 25 57 [1]
336 0.023962359873140659995666055511 29.2124817299246047612546879343 0.865865194560796187746648521905 673 50 C2 [2]
337 0.023844924266386995495058727568 29.3563524119367913393239397230 0.859950846593129688658502594937 587 18 56 [1]
338 0.023808208581619536230919138724 29.4016241331326157454440956567 0.859848564104030812964031942750 608 28 57 [1]
339 0.023785909384304898211246549766 29.4291880411305233453007822820 0.860777786592273733508471747749 559 19 57 [1]
340 0.023781994142671214607833421901 29.4340329831304626302464960538 0.863032767677865722592090411766 907 55 [2]
341 0.023717909889284751816076319841 29.5135618301781785502919029877 0.860912554585589902697711197079 780 2 56 [1]
342 0.023677381273894284292880212638 29.5640802461457793919278439154 0.860488902914367367755107909530 553 14 49 [1]
343 0.023668784562508404000144620316 29.5748181809389204973769455150 0.862378391691097541123822683754 628 10 54 [1]
344 0.023649449168948689160548290579 29.5989980569647976919116200081 0.863480103478617719503738308932 713 10 58 [1]
345 0.023644204643894152456048721616 29.6055634157593478903257604113 0.865606176964140913812957459388 690 29 61 [1]
346 0.023638241609579585311297096498 29.6130317796700849246120886582 0.867677360798112945165625557250 454 52 59 [1]
347 0.023634338212668059164282292883 29.6179226048647473496185925617 0.869897734196426000552485857389 563 31 57 [1]
348 0.023625447345592903435571749019 29.6290685954176510614972170597 0.871748397683273186441534437609 661 5 56 [1]
349 0.023621591172422675888258728595 29.6339054761570768362415931325 0.873968051771953408374542711522 889 56 D1 [1]
350 0.023470511529669562202069590188 29.8246588752492858259729214811 0.865296579488399825823426856378 869 5 58 [1]
351 0.023436328093724970764003343767 29.8681601145285032062721290293 0.865242986354018469974867921967 649 36 60 [1]
352 0.023420082202713330611177076439 29.8888788664841488061816008859 0.866505505280438534022987354966 556 13 43 [1]
353 0.023373728664177613049121185897 29.9481529052236463168585932031 0.865530815046942812477972651034 789 10 59 [1]
354 0.023336629330658799271231022397 29.9957628876748677084798710827 0.865229565019133890130823006775 802 5 57 [1]
355 0.023308684020541875472291449299 30.0317254883669981025065170996 0.865596904573846337708042824916 730 6 54 [1]
356 0.023291928286393933536967310679 30.0533296939999425145060700326 0.866787658717298397283363683651 676 8 47 [1]
357 0.023279000699912890278682964543 30.0700192857771335089990322340 0.868257844085810686567089763862 853 1 39 [1]
358 0.023156626393226301161916476978 30.2289283470394313059717139124 0.861559820298125571100487510626 876 8 50 [1]
359 0.023078899867494748405653375531 30.3307351745092571716957043882 0.858176241609700164278817632228 560 45 57 [1]
360 0.023078888404996200187993000107 30.3307502387533318093072398897 0.860565849850489569425536725737 583 45 59 [1]
361 0.023078710080622639973623769858 30.3309845981268428217908153027 0.862942974932658309473336500900 79 293 67 [1]
362 0.023065112748270831154721259735 30.3488653031829776063776204593 0.864314038452203289596517782241 719 3 53 [1]
363 0.023050176900742576256967081222 30.3685304895620584282574561931 0.865579541881202063063063645526 813 52 D1 [2]
364 0.022928303185165924118968541968 30.5299521882144173409323666373 0.858809911598603247486468546891 879 5 53 [1]
365 0.022916514490583086467920315139 30.5456573811713737854429135934 0.860283958696708773679754949489 275 161 54 [1]
366 0.022896524634503335895574445483 30.5723253277116471252229673911 0.861136610634620999217773571842 308 135 58 [1]
367 0.022780601122152370504659895908 30.7278985416808956228589727039 0.854768002926632958746233773564 504 48 48 [1]
368 0.022738614896548940637530613647 30.7846367593058453577320431521 0.853940604005919195437486274660 649 8 56 [1]
369 0.022736795760008753135895737941 30.7870997913969262101376353955 0.856124094916892929864111107022 632 16 61 [1]
370 0.022698489079468187873697177580 30.8390570645154406246785318418 0.855554058104706436321388210602 519 37 45 [1]
371 0.022664578348515222597425867978 30.8851984464938278777805598670 0.855305037721244292973097867458 677 7 57 [1]
372 0.022659857568601391825628387801 30.8916328304708444024162057810 0.857253217884714579499012992974 715 7 57 [1]
373 0.022646912489741795661452021726 30.9092906292225847224928116760 0.858575849722043655983855405151 613 24 56 [1]
374 0.022636229213064692303082973491 30.9238784168163946522658977524 0.860065645886548044560245873323 625 18 61 [1]
375 0.022629688520526817506843113387 30.9328163913987469584953668936 0.861867001248236120311487802226 553 42 53 [1]
376 0.022565664523671938285815923662 31.0205799286647528743692809191 0.859282431092210638932576469114 649 17 53 [1]
377 0.022541468199915560058504828206 31.0538778482327157211837240432 0.859721093095586965043717094701 739 15 60 [1]
378 0.022529892709989323399998455554 31.0698328221347184647625776119 0.861116438037664539674716677284 666 11 60 [1]
379 0.022490863284948034279736335357 31.1237497258930747207115073007 0.860405729873509477709099823938 591 37 57 [1]
380 0.022477385189315918971604957092 31.1424124338416399903517852310 0.861642288195134156006537876602 648 22 60 [1]
381 0.022444635197571134566532951885 31.1878537493784305265024286713 0.861394134897073856727494563347 625 60 65 [1]
382 0.022425460879602404526399912175 31.2145201277312946762227166323 0.862180011552741535179178731613 481 92 63 [1]
383 0.022417629969157757106164870344 31.2254239615455385018383775505 0.863833414542085938092010442736 505 79 59 [1]
384 0.022415915304074329999081716168 31.2278124941330228326413687154 0.865956369653000314872032436592 515 47 56 [1]
385 0.022405618826067047973139253650 31.2421632017415664072423086150 0.867414042801777627120602035685 628 23 56 [1]
386 0.022387690233130496222859652860 31.2671826664861890383187679205 0.868275837709435555018016629209 796 15 60 [1]
387 0.022382982114197555619841001823 31.2737595208990968080389516735 0.870159153513099030376184786330 1060 1 59 [1]
388 0.022309954169559146542689814034 31.3761289996335422748427292597 0.866724183575044274509438797736 626 40 63 [1]
389 0.022287534410154598132169998561 31.4076912734262573638066381807 0.867212416622560172722566095586 690 15 62 [1]
390 0.022256839566856413886855711117 31.4510062355123920429270180352 0.867048578007036919045957732843 367 161 58 [1]
391 0.022254755643108566197068935618 31.4539512913844472233148956055 0.869109006037526615455284601029 431 90 53 [1]
392 0.022243987272294359801460576518 31.4691782292050548582207309187 0.870488775555053381044103295265 363 117 46 [1]
393 0.022243834742772965701797817097 31.4693940183776266326764418816 0.872697441694534964432324143051 194 247 56 [1]
394 0.022234713903364975509413541468 31.4823029899234647430732549054 0.874200691959572265704915616402 883 12 63 [1]
395 0.022227010524294292443884629539 31.4932140440071618275897026983 0.875812296780520519583673950279 807 1 53 [1]
396 0.022224820450688262909166183136 31.4963174417151366876191589330 0.877856523469405191620958302261 814 60 C2 [1]
397 0.022050168654090634072705368773 31.7457889316479034790188835466 0.866295723735001124443317015866 724 18 54 [1]
398 0.021987828834989929408614308710 31.8357944867238459780248984929 0.863574082404314674555822844064 788 14 65 [1]
399 0.021964481426075379747719034230 31.8696347262260957407608456041 0.863906291310022274061689999363 822 19 66 [1]
400 0.021933319714099063129790051849 31.9149134341952632327470410240 0.863615766650595138080208958971 725 12 58 [1]
401 0.021888084211858707870673848487 31.9808711088907539961853457697 0.862207323094069170914726837407 754 5 51 [1]
402 0.021836340064251285358244249227 32.0566540885660678724096783205 0.860275557945604112939783490013 732 23 56 [1]
403 0.021811109086428706270625581658 32.0937370596873284408231226996 0.860423728090291276778546107903 922 12 61 [1]
404 0.021793869612576501436229101973 32.1191239758566689408483025450 0.861195782634995265371366624522 691 15 53 [1]
405 0.021769834235195380018014420043 32.1545856728806477506918125788 0.861424263387103983332971786245 780 3 48 [1]
406 0.021754387226292455348588536777 32.1774174891019999604904428582 0.862326188400472026485645600328 674 13 45 [1]
407 0.021713851777567236055874943598 32.2374863368633620842347542904 0.861231646310092951752531398777 722 7 60 [1]
408 0.021639434169714228251899616294 32.3483504471523917233482158047 0.857440113084054754710746637048 857 17 65 [1]
409 0.021612675227558085671702790896 32.3884013723316241855977578407 0.857417208214686595067594220880 814 13 69 [1]
410 0.021579777953391747942506696045 32.4377758432856924055626042959 0.856898993535280501220035127377 745 17 62 [1]
411 0.021516223073151526773268893588 32.5335909383407186030462064328 0.853936802741316765397703988483 941 16 61 [1]
412 0.021489895834497482408345310705 32.5734477910450388432906851202 0.853920951721742861025769381311 912 9 64 [1]
413 0.021468508894633577352886698028 32.6058974768842481952440628509 0.854290638012971909023640011400 757 32 64 [1]
414 0.021468235524846591371421636042 32.6063126701700413597324793936 0.856337329545619058416818288644 662 38 57 [1]
415 0.021455213556040363560052394100 32.6261026566629759043421105387 0.857364728151582262280394301592 612 52 67 [1]
416 0.021424571098572988306917036953 32.6727660861609696004750443442 0.856977532808549265519225796524 803 15 64 [1]
417 0.021409775909464243861879223297 32.6953445454122332067032951764 0.857851531615477840790199508455 792 14 66 [1]
418 0.021384194087312495721924082476 32.7344578496562830024073664016 0.857855005110709586799229779189 856 16 66 [1]
419 0.021363349107804896245247360825 32.7663980243744489991636808140 0.858231658420085452972840224688 788 14 65 [1]
420 0.021347084766781176812433908193 32.7913627386391644864875142048 0.858970546287757485732907943258 810 7 62 [1]
421 0.021338099914628065128558515021 32.8051702260576553669513916878 0.860291074713814840441661881313 745 13 64 [1]
422 0.021329295871177340118233467011 32.8187111392609330777234200480 0.861623074373519650439832075243 740 20 65 [1]
423 0.021319302329004406793595676464 32.8340950936122590280670735496 0.862855708740219994524512511235 672 26 60 [1]
424 0.021309657165625517578882045365 32.8489564407054786279293075911 0.864113151043466899225118382726 815 22 63 [1]
425 0.021276042541124347957403537353 32.9008554409013693572072302788 0.863420712647048133997060434782 1014 16 61 [1]
426 0.021261229008882521614911795395 32.9237787574534766330933023843 0.864247560896151602946374311307 774 19 64 [1]
427 0.021243080304677384049374930506 32.9519066896278350580562318686 0.864798025619446892972344233730 911 16 65 [1]
428 0.021221184358451038964220265486 32.9859063554685542383412901724 0.865037307678248033043558614503 522 88 58 [1]
429 0.021209402920442373202881020456 33.0042294271902970345227333341 0.866095954503952220085353172232 618 67 63 [1]
430 0.021200783252711678352811766209 33.0176480583785393164348053851 0.867409352225857419597324086151 1177 6 66 [1]
431 0.021183838670058295663819910390 33.0440582985271419972621771103 0.868037372023229509152982302670 856 3 66 [1]
432 0.021162400852159088515975663451 33.0775324071315225863397009639 0.868291305526067655120463737639 730 17 64 [1]
433 0.021156769462640853172050747495 33.0863367980672717750210184097 0.869838120271134126886486165961 557 62 60 [1]
434 0.021154550142252941432823247146 33.0898078802374665303306623638 0.871664082424617244199797536686 700 32 63 [1]
435 0.021151709002469172968128228953 33.0942525692975714707252377777 0.873437865416035452235869446884 714 24 62 [1]
436 0.021146168351962472767012467129 33.1029238181127227003498920862 0.874987185870586272526780015135 554 75 58 [1]
437 0.021143440048726568753888364984 33.1071953469634066016586991233 0.876767750231744351761667874992 1130 63 D1 [1]





Updates

Please note that the results are taken from a running search. For updates look at the list below.

19-Aug-2010: First complete presentation from N=1 to N=437 by Eckard Specht [1].
20-Aug-2010: First improvements for N=23, 31, 36, 37, 38, 44, 60, 336 and 363 and a forgotten packing for N=7 by David W. Cantrell [2].
23-Aug-2010: Further improvements for N=46, 47, 48, 50, 61, 150, 175, 195, 312, 313 and 340 by David W. Cantrell [2].

References

[1]   , program crc, 1999–2010.
[2]   , private communication, July 2010.


©  E. Specht     23-Aug-2010