a remarkable coincidence showing that numerical data can be misleading!

Thu Jun 7 06:37:58 CEST 2007

Interestingly, there were discussions about exactly this problem a
couple of months ago:
http://lib.mexmat.ru/forum/viewtopic.php?t=6838 (in Russian)
http://www.mathlinks.ro/Forum/viewtopic.php?t=140737 (in English)

There I showed that all elements of A129935 must be halved
denominators of some convergents of 1/ln(2), and (re-)discovered the
smallest such element.
Namely, if n belongs to A129935 and m=ceiling(2/(2^{1/n}-1)) then
0 < m/(2n) - 1/ln(2) < ln(2)/3 * 1/(2n)^2,
implying that m/(2n) is a convergent of 1/ln(2).

Anyway, several first terms of A129935 are:

777451915729368, 140894092055857794, 1526223088619171207,
54545811706258836911039145, 624965662836733496131286135873807507,
1667672249427111806462471627630318921648499,
36465374036664559522628534720215805439659141,
123447463532804139472316739803506251988903644272

First 56 terms of A129935 are listed in the attached b129935.txt file.

Regards,
Max

On 6/6/07, N. J. A. Sloane <njas at research.att.com> wrote:
>
> Dear Seqfans, When I was up at MIT recently Richard Stanley showed me a
> remarkable coincidence, which is now encapsulated in the following sequence:
>
> %I A129935
> %S A129935 777451915729368
> %N A129935 Numbers n such that ceiling( 2/(2^{1/n}-1) ) is not equal to floor( 2n/(log 2) ).
> %C A129935 In latex: when is $ \left\lceil \frac{2}{2^{1/n}-1}\right\rceil $ different from $ \left\lfloor \frac{2n}{\log 2} \right\rfloor $?
> %D A129935 S. W. Golomb and A. W. Hales, Hypercube Tic-Tac-Toe, in More Games of No Chance, ed. R. J. Nowakowski, MSRI Publications 42, Cambridge University Press, 2002, pp. 167-182. Here it is stated that the first counterexample is at n=6847196937, an error due to faulty multiprecision arithmetic.  The correct value was found by J. Buhler in 2004 and is reported in S. Golomb, Martin Gardner and Tictacktoe (unpublished).
> %O A129935 1,1
> %K A129935 nonn,bref,more
> %A A129935 Richard Stanley (rstan(AT)math.mit.edu), Apr 30 2007
>
> The companion entry was already in the OEIS, but has now been revised:
>
> %I A078608
> %S A078608 2,5,8,11,14,17,20,23,25,28,31,34,37,40,43,46,49,51,54,57,60,63,66,69,72,
> %T A078608 75,77,80,83,86,89,92,95,98,100,103,106,109,112,115,118,121,124,126,129,
> %U A078608 132,135,138,141,144,147,150,152,155,158,161,164,167,170,173,176,178,181
> %N A078608 a(n) = ceiling( 2/(2^(1/n)-1)).
> %C A078608 For n >= 2, a(n) = least positive integer x such that 2*x^n>(x+2)^n.  For example, a(2)=5 as 4^2=16, 5^2=25, 6^2=36 and 7^2=49.
> %C A078608 Coincides with floor( 2*n/(log 2) ) for all n from 1 to 777451915729367 but differs at 777451915729368. See A129935.
> %o A078608 (PARI) for (n=2,50, x=2; while (2*x^n<=((x+2)^n),x++); print1(x","))
> %Y A078608 Cf. A078607, A078609, A129935.
> %O A078608 1,1
> %K A078608 nonn
> %A A078608 Jon Perry (perry(AT)globalnet.co.uk), Dec 09 2002
> %E A078608 Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Dec 17 2002
> %E A078608 Revised by njas, Jun 07 2007
>
> Maybe some seqfan can extend the first sequence!
>
> Thanks to Andrew Plewe we have had a lot of discussion recently
> about possible duplicates. This example shows once again that
> numerical data can be misleading.
>
> Neil
>
-------------- next part --------------
1 777451915729368
2 140894092055857794
3 1526223088619171207
4 54545811706258836911039145
5 624965662836733496131286135873807507
6 1667672249427111806462471627630318921648499
7 36465374036664559522628534720215805439659141
8 123447463532804139472316739803506251988903644272
9 5126378297284476009502432193466392279080801593096986305822277185206388903158084832387
10 1868266384496708840693682923003493054768730136715216748598418855972395912786276854715767
11 726011811269636138610858097839553470902342131901683076550627061487326331082639308139922553824778693815
12 6713433343972698590657018335263249071454687710830959488373107240423034186603602852451709579990096719886095110926
13 18868483929561879615785894693105827328040410884985641488001039423069804706763264210393431426768050160588860286036972089918385413161
14 37313209045635664493729157679587661880114398048696578780116355222319962748263488263660429837731699039055106149095537460331879854298435
15 60452902470369776601717524797749530523935048629781736499744444103082801695293031232358571112626089581150538477607351214658927268748405227
16 56380427908963491263909511136946875875982758156157202257620228271869992177202279519199104184022144459431369747323288146217498124081377476276000819182718
17 1812394161112545037283369947319723920466124728340212692621746696556231011594115430822909956760315689740127810278223104111671703497328171081570023330159887746285
18 2054557677180967662133738282246364228482520430709973792245484826151744589581682084661721675969957309063692987703323786258988253250345199116998712144254069907758779
19 808602740249290901914306249215825460316348565269854670233597277844441117525124008735123544322202902832280461810398307612748374928982977976112555721931824524461419110
20 1034512570271189908345576814661951885091138692596020056443287154414555396854748421337002459920268323244978688671338164491409226954348996802588271512654000176632887057511979818150883239274
21 26394938909807794985370408357326580977553440763181131653001236847702753477807396570276184104893865012802515111641707157138391873724176694557985708140326124111306281841805577848750772265620363385243046909
22 1285463759714102522366947036385200502859567355201553177267777450631129638244453256730870551739941264232921625957863651199027994455588234743208670313800418734772713907524885964503224237634416042221930882177153009288246563
23 4073773987912141862632061930801889069594176689646265554103928277214068113297516930555211543445500470714410923227954648740991531281191868009881492784972873394405484929055224865741474922052407081092647754970970162355617228470
24 384065706689305388232542362482282603627458050007721393869459493661053356705814302310248356864743910610628336618304398970281146073524311616442356516311292592534159586744521702428380713587057775511374477676037508418515335690487
25 126090710299020836758051056849867316486829808614696134278994923276639947936773569213383396606475062649234398760548003116179000316723732469083880705541447806886330649687973811697676357027958496540685039776285726010312920239100645
26 314021040652724263185467389966163664917075275464875564133262532500851738742722104932733989157110416277750752232244126520664909525305948465865267828842391381552259532776370515338163185373502100950178108386047241859761748291098501389
27 165018006789761363152011110795012800158778136004492117659587752199909189277445090116418532102856737898237788597923294666833840991546911715751037387593159723068180102054657917905378500814891092802482861678649544346640408455529271113650274
28 7382159663869533628243456893924382981217202943443694806064591326850421926676110439059470577242128549902692608519079890794798331581953333342091173182955169225318203329287201771670224645683234392848950509882468250205609140642118420938105506247410
29 2334095202510641893405626962259305909638849193888082953189512521032712327121135333084986464579731525766515539207025541141822385768572090240390061939868062673202727474939550308024602505073125318544432839017093911163527947121750519961947757315382426
30 25774134436592688496070602680188986828325096659904116233817709831105408970736595962054630463120242486651294508336177986801835483383709982745901665889972723551184579741076218619562498397944511646086938746191960418400166393009247734565681695982728775696372088
31 102460983970901332413682137071980393345402826693320192090360813129349742500097775916705911973201894400867228048556155968286501343641333007798882756764020967596194513108746210969589217482707465349421544946843772623951965344605825672336079682516213959349753502289704
32 2524441132445024310795337963700417024813804517390919110733110433183106143324916741275249456033500022672588829852225770097245849930331802950485638538540462976840547656548819997191490063909656238277234872075795558957233717485859413420656814851861105517722106998289037385889737399085305755665497871568
33 19305689670995098750872359531556267769799271065457368391164246186742344079923795287564311294115946563552911599278715992422095028869981018006629492017640689800492679950773626287996598789324985306170123763699494105964158157525435022229783112679266665637096650131616656407730448094922345359759979657416252
34 825938178725862470668731143853500055770827366427774395569242416121947420931438276432101429051717195553612323601798832846355583992322952343538822134069965039223614070919286865817634667434475142051826747655585028110155598130153493113423366879864589933523243147286414596703368678554144766303539160428257852463856280045877280683594018
35 4390856652776138614633923445972808289738477585363695842903568565093196504382890249941848743423399643132690228236425497586860432289893304839226372445536895496895883166724175714142234236277194632015700980736988583834698565258198637423870154300203368798346081221627341471664420664391349633231084864890578226357724264146220709009928225558581100966
36 18093351785914680782862059174484708448168344817942108945877040899328862193388544126898546383978633102620263674988926058769190003882974504908312603486071313453867897626044307783512756575362862592464085632301398604723693152135733428161538975605252992408501164700941631654558621796513667570880110085869956009933560490633651947369252961947242351548037831
37 5498133957906599987697686679715729443086772561800102567639225681399554074885706482720438680920261939452379122993270116714472842065362207021142005177042062044217228789691533785595627983727752047929709625778261258354777532431740748411357662458587755253872107955201990147962204241406749675451179406818208122312134698754074604200103810811769322132992653179994507
38 85241416533883091940906274129898039036238761336570017182506794818495232310288387090549502685755474869012122625144971434192308813179066100184711629847261692399440179824187778099589676445787482709469537320469474269989880085585021607659984130687020095666325932806797090488781315474963651566723311676680642431488403761612082608631991200953641667712669873188456686476426
39 1154164016065716659036627435473882381141458778656913923027758942834865923295759796073797401840167612404347166628450162132860201093386284125641817501448477827501596724495944248807402613528920205308421927668103851716007242232161707912993685674587867198736312654658468222469696869661781727543470407260186319270064030371140561813315406825599154756881528216433339711716541
40 301328859395402520936169452402253717225261450992017541309555425300288861786717328303054429614134439715748989635028623383653232858417937006438995500504920864630534493867004455205757555770515539445668759006379106670290167824833318619318179745018746550780325753922200418156821962518699539403197911005835305264808141489808053609077520223907783017699312504219068470333063207
41 574804826684178390961092489624871498752778120818377282901595594528210977021356358957733434844488275180058535779987846692212998346333722741622382279596152203121341026113630028953780467998642256021240636249327696852034985272848446926194689933154302959249257928426188207233850135637007502664528892529134544691923893035532833994981885582741142184186348603201914313691900153968
42 34318259929560487572873060828760854566171233142299962452812654746244860992085668433134067860395314675359285495428309042447850216579960803239285249679240664357012761796587355277707810997408752693125088371631888470976755231139905171645983286674959298158752130102754874952994621094048235619672962854075455093068229906268418648908644741903645156163857196221449266189663146766444
43 327685773670063689043022962619564080267426844045265397439992112592923507108477276275339592610211693070983881610781943195444614908625253591908168173004683260531114625440379234317735805784959037707471869434895226227342554126283810639561069747050186757040141785515028104670572394100064956767464744049645993980197958731377666133706066321706880916868732312272612880249390712570027489684
44 16909018430237712198232679828018232199911652618381858007269470911300333225000302301547721154203531669317496539372544753376032707564933182084789835979694959315687692332521239781021007610782744455007958240869904714510473711412953666814658959025225903866737868412042919342481591703611614798746170840567008065609386333455504941714706915225384432103072584611650341569196793931697953337533196
45 139913012397089952910589659243662194237720314247381212796941757572028328634301795852849708382825404794825472670748973655850753913221503046531154689608338534630846711099054276416914773563385403572510300185964355100193625103195789025532408135235399232713504097918791640791853534098850496484921137977589918615223011874383149224620760459597949078096411587903040927862880165808259759866243941637879
46 708247228835634647683418459791820631303326501128391901750445166215927616965020361298538292973544262270977787521723653419743450523207377285950605521537084195272595809911357767293902933078158998507240855878148137213907254696087212159753535442074361451193101355173210777821836199255888380026285219951156322966756497513292384001189891545337860616794780913489251146499118951532903566900184196592880492971140627462
47 8342467286243225771876940720347578585106375994632075295630463755279081891333886282949272472713514908767087005267301851630526136374772556894638402886758551256937568365175135845253465197968810098105034615010807661536083614296502117834191092242903046468494430066917669208460404063737967363316684491933469899303748703443544332480134052114093822171354096193894319282436678614583135328791855956637324294036868545476180139808577
48 72589409469730448004222155988992659413162360044870871825763607085164730395001919481363698878290780201425882459969612015944584447991571761917148877856508129452342734533192702452911260981188301181130238678497811516540584087671570547561375473185853313418114447173407559901155542226067178305596363620934728532779953139482286406194327029348522966935265605151593334994885736730425482662577017349226483200525608074279473314338688
49 21308810867972697693276133735424284948956046884553370293368441376091172570913248850381328594152911051473366002304928505909794875922172202869433367944460780714655185091598201316138447202669138640186305747934872714138541833815002968475911413344248129940341406263318287810985457199825469781995105013328731605362980961858967237485914879433986140685063175705806151653901809916614110444483445093811006326799323575260046340457323832157798
50 42783898145131954133858805210497482267121004915541506154465808462541538848091714692553237787328538501421317496751372253459204325715024598259410561985448247982392238904714753221741298433419171247464691334837994415480317731316564303089455385238898191820789067855986378356115914975020491380087311664017467614621676780181246933033585250692433748773818036788480484193771564330169727020354862098449515732246089304620120044854020437464769572456537
51 1831818673613185754746818686991356738383131544534391214519013360203914537826448097573002092379365404162734919520232133406751220902099285162717225499101760363005876288081275139764966269605366921676849637610371743088864660351246712179023272180042737993374757799181582957330480301489538091633438428728591165926415554629068750707981384841008978928747779466355590666307261127817315487352866411895514560129813271575627621019973501880548993049649066901508975
52 1361215655731706027082824002543860280748193350087161180447927532795767400412483442268974401721416848586913194687374583707945414556477172123799179056212426352666341257380183763739719486296075132564333593889071121064647957089124910559345868585458197569856355845728908790152991125469150480017030468691844928241992081485480131766422383790904953520916055012817551082123909414175505869879635722249479453635021777048898993799594092548049288679866031124962472723
53 454109248048932354903464276997836026734613952307559375696089718624653534334892418863302283910400708040264011248930949965958562704131189175945038542272474775018914183030858605826333531844070727142224166756880915585629268175445109586093710238318132488566035718211945363466472338153380061379587481289333137123259555699976050368817041549961177238447348244266678748115621813870472284751296371289073579849658410230495264349711292821874237368513881439848943525373157622692
54 469203442807533123011900580547488106806662755988473710355064870795132297572573911530940691352711308322064227568278118971672651681633104487766523712893080962916695533534780195877447012802097168688645749093673592313319504494627959497050460612645398867239895715152343887321356387767855532112330750608674762584686057940809066949787180856057565044970356028274752082770925251178396014412918417116948952680951472662093397524176971307066175466101447200201694685211640478933611
55 29585154698688612922217682364560425878591641634951474355124728415827200025128277454509375846521828689566480064561360910784117958273638500204890484651418198419867584916439895547699249410817000173220439140194783165003871154795423746389177092483276727594748637064296807168378311343722449990755267080947990336318873693426923060587912305489167549393550370791040758803974929182582977315696463906516198066418921254157930687388545260454047124701079440278585783857073148406401867
56 8174076117870717802035867683835265325510299227436002588372666186968000764739536900911883880496519089449406662006738188040667431900460979304863476233292186847891273500304986600525494889204964424826062744235818806388805220319134939865297934744252432527288910861095557992067295430350556728814994551137349733802771213978560605703184556465948566897839341471105944485599590412379808648169550642318170653400514993412114818750233197324245321836613533265402996763794960537089736527848453