Max even and odd proper divisor
Consider
the hereunder table where the largest even and
odd proper divisors of N are highlighted:
Idea A
We subtract ev
from N then add od to the result and iterate:
1,1,1,1,1,1,1,1,1,... loop
2,3,4,3,4,3,4,3,4,... loop
3,4,3,4,3,4,3,4,3,... loop
4,3,4,3,4,3,4,3,4,... loop
5,6,7,8,5,6,7,8,5,... loop
6,7,8,5,6,7,8,5,6,... loop
7,8,5,6,7,8,5,6,7,... loop
8,5,6,7,8,5,6,7,8,... loop
9,12,9,12,9,12,9,12,... loop
10,13,14,19,20,15,20,... loop
11,12,9,12,9,12,9,12,... loop
12,9,12,9,12,9,12,9,... loop
13,14,19,20,15,20,15,... loop
14,19,20,15,20,15,20,... loop
15,20,15,20,15,20,15,... loop
16,9,12,9,12,9,12,9,... loop
17,18,21,28,21,28,21,... loop
18,21,28,21,28,21,28,... loop
19,20,15,20,15,20,15,... loop
20,15,20,15,20,15,20,... loop
21,28,21,28,21,28,21,... loop
22,31,32,17,18,21,28,... loop
23,24,15,20,15,20,15,... loop
24,15,20,15,20,15,24,... loop
25,30,39,52,26,37,38,55,66,77,88,55,... loop
26,37,38,55,66,77,88,55,66,77,88,55,... loop
27,36,27,36,27,36,27,... loop
28,21,28,21,28,21,28,... loop
29,30,39,52,26,37,38,55,66,77,88,55,... loop
30,39,52,26,37,38,55,66,77,88,55,88,... loop
31,32,17,18,21,28,21,... loop
32,17,18,21,28,21,28,... loop
33,44,33,44,33,44,33,... loop
Question
Are we doomed to loop forever?
____________________
23.00 update#1
Giorgos Kalogeropoulos was quick to send this:
GK
> Some quick results on record-length chains
5->{5,6,7,8}
10->{10,13,14,19,20,15}
22->{22,31,32,17,18,21,28}
122->{122,181,182,247,266,361,380,285}
134->{134,199,200,125,150,175,210,245,294,343,392}
2917->{2917,2918,4375,5250,6125,7350,8575,10290,12005,14406,16807,19208}
3334->{3334,4999,5000,3125,3750,4375,5250,6125,7350,8575,10290,12005,14406,16807,19208}
78125->{78125,93750,109375,131250,153125,183750,214375,257250,300125,360150,420175,504210,588245,705894,823543,941192}
125000->{125000,78125,93750,109375,131250,153125,183750,214375,257250,300125,360150,420175,504210,588245,705894,823543,941192}
390625->{390625,468750,546875,656250,765625,918750,1071875,1286250,1500625,1800750,2100875,2521050,2941225,3529470,4117715,4941258,5764801,6588344}
625000->{625000,390625,468750,546875,656250,765625,918750,1071875,1286250,1500625,1800750,2100875,2521050,2941225,3529470,4117715,4941258,5764801,6588344}
Midnight update#2
> I will just leave here some more records below 100000000
1953125->{1953125,2343750,2734375,3281250,3828125,4593750,5359375,6431250,7503125,9003750,10504375,12605250,14706125,17647350,20588575,24706290,28824005,34588806,40353607,46118408}
3124999->{3124999,3125000,1953125,2343750,2734375,3281250,3828125,4593750,5359375,6431250,7503125,9003750,10504375,12605250,14706125,17647350,20588575,24706290,28824005,34588806,40353607,46118408}
6510418->{6510418,9765625,11718750,13671875,16406250,19140625,22968750,26796875,32156250,37515625,45018750,52521875,63026250,73530625,88236750,102942875,123531450,144120025,172944030,201768035,242121642,282475249,322828856}
48828125->{48828125,58593750,68359375,82031250,95703125,114843750,133984375,160781250,187578125,225093750,262609375,315131250,367653125,441183750,514714375,617657250,720600125,864720150,1008840175,1210608210,1412376245,1694851494,1977326743,2259801992}
78124999->{78124999,78125000,48828125,58593750,68359375,82031250,95703125,114843750,133984375,160781250,187578125,225093750,262609375,315131250,367653125,441183750,514714375,617657250,720600125,864720150,1008840175,1210608210,1412376245,1694851494,1977326743,2259801992}
Merci Giorgos! Love those beautiful chain-records!
___________________
Next day update#2
Jean-Marc Falcoz has sent the hereafter interesting "trick" to produce quantities of terms arbitrarily as large as one wants before entering a loop: "Start with 5^k. For these terms, we obtain a sequence containing 2*k+2 terms before looping. For example for k = 3, 5^3 = 125 gives 2*3+2 terms : {125,150,175,210,245,294,343,392}, then 245..."
JMF
> Un truc simple qui donne des records de longueurs arbitrairement grands "avant looping", c'est de commencer par 5^k. Pour ces termes, on obtient une suite contenant 2*k+2 termes avant de boucler. Par exemple pour k = 3, 5^3 = 125 donne 2*3+2 termes : {125,150,175,210,245,294,343,392}, puis 245...
Pour k = 10, 5^10 = 9765625 donne 2*10+2 termes :
{9765625, 11718750,13671875, 16406250, 19140625, 22968750, 26796875, 32156250, 37515625, 45018750, 52521875, 63026250, 73530625, 88236750, 102942875, 123531450, 144120025, 172944030, 201768035, 242121642, 282475249, 322828856}, puis 201768035
Par exemple pour k = 100, on obtient 202 termes :
{7888609052210118054117285652827862296732064351090230047702789306640625, 9466330862652141664940742783393434756078477221308276057243347167968750, 11044052673094165275764199913959007215424890091526322066783905029296875, 13252863207712998330917039896750808658509868109831586480140686035156250, 15461673742331831386069879879542610101594846128136850893497467041015625, 18554008490798197663283855855451132121913815353764221072196960449218750, 21646343239264563940497831831359654142232784579391591250896453857421875, 25975611887117476728597398197631584970679341495269909501075744628906250, 30304880534970389516696964563903515799125898411148227751255035400390625, 36365856641964467420036357476684218958951078093377873301506042480468750, 42426832748958545323375750389464922118776257775607518851757049560546875, 50912199298750254388050900467357906542531509330729022622108459472656250, 59397565848541963452726050545250890966286760885850526392459869384765625, 71277079018250356143271260654301069159544113063020631670951843261718750, 83156592187958748833816470763351247352801465240190736949443817138671875, 99787910625550498600579764916021496823361758288228884339332580566406250, 116419229063142248367343059068691746293922051336267031729221343994140625, 139703074875770698040811670882430095552706461603520438075065612792968750, 162986920688399147714280282696168444811490871870773844420909881591796875, 195584304826078977257136339235402133773789046244928613305091857910156250, 228181688963758806799992395774635822736087220619083382189273834228515625, 273818026756510568159990874929562987283304664742900058627128601074218750, 319454364549262329519989354084490151830522108866716735064983367919921875, 383345237459114795423987224901388182196626530640060082077980041503906250, 447236110368967261327985095718286212562730952413403429090976715087890625, 536683332442760713593582114861943455075277142896084114909172058105468750, 626130554516554165859179134005600697587823333378764800727367401123046875, 751356665419864999031014960806720837105388000054517760872840881347656250, 876582776323175832202850787607840976622952666730270721018314361572265625, 1051899331587810998643420945129409171947543200076324865221977233886718750, 1227215886852446165083991102650977367272133733422379009425640106201171875, 1472659064222935398100789323181172840726560480106854811310768127441406250, 1718102241593424631117587543711368314180987226791330613195896148681640625, 2061722689912109557341105052453641977017184672149596735835075378417968750, 2405343138230794483564622561195915639853382117507862858474254608154296875, 2886411765876953380277547073435098767824058541009435430169105529785156250, 3367480393523112276990471585674281895794734964511008001863956451416015625, 4040976472227734732388565902809138274953681957413209602236747741699218750, 4714472550932357187786660219943994654112628950315411202609539031982421875, 5657367061118828625343992263932793584935154740378493443131446838378906250, 6600261571305300062901324307921592515757680530441575683653354644775390625, 7920313885566360075481589169505911018909216636529890820384025573730468750,9240366199827420088061854031090229522060752742618205957114696502685546875,1, 088439439792904105674224837308275426472903291141847148537635803222656250, 12936512679758388123286595643526321330885053839665488339960575103759765625, 15523815215710065747943914772231585597062064607598586007952690124511718750, 18111117751661743372601233900936849863239075375531683675944805145263671875, 21733341301994092047121480681124219835886890450638020411133766174316406250, 25355564852326440721641727461311589808534705525744357146322727203369140625, 30426677822791728865970072953573907770241646630893228575587272644042968750, 35497790793257017010298418445836225731948587736042100004851818084716796875, 42597348951908420412358102135003470878338305283250520005822181701660156250, 49696907110559823814417785824170716024728022830458940006792545318603515625, 59636288532671788577301342989004859229673627396550728008151054382324218750, 69575669954783753340184900153839002434619231962642516009509563446044921875, 83490803945740504008221880184606802921543078355171019211411476135253906250, 97405937936697254676258860215374603408466924747699522413313388824462890625, 116887125524036705611510632258449524090160309697239426895976066589355468750, 136368313111376156546762404301524444771853694646779331378638744354248046875, 163641975733651387856114885161829333726224433576135197654366493225097656250, 190915638355926619165467366022134222680595172505491063930094242095947265625, 229098766027111942998560839226561067216714207006589276716113090515136718750, 267281893698297266831654312430987911752833241507687489502131938934326171875, 320738272437956720197985174917185494103399889809224987402558326721191406250, 374194651177616173564316037403383076453966538110762485302984714508056640625, 449033581413139408277179244884059691744759845732914982363581657409667968750, 523872511648662642990042452364736307035553153355067479424178600311279296875, 628647013978395171588050942837683568442663784026080975309014320373535156250, 733421516308127700186059433310630829849774414697094471193850040435791015625, 880105819569753240223271319972756995819729297636513365432620048522949218750, 1026790122831378780260483206634883161789684180575932259671390056610107421875, 1232148147397654536312579847961859794147621016691118711605668067932128906250, 1437506171963930292364676489288836426505557852806305163539946079254150390625, 1725007406356716350837611787146603711806669423367566196247935295104980468750, 2012508640749502409310547085004370997107780993928827228955924510955810546875, 2415010368899402891172656502005245196529337192714592674747109413146972656250, 2817512097049303373034765919006119395950893391500358120538294315338134765625, 3381014516459164047641719102807343275141072069800429744645953178405761718750, 3944516935869024722248672286608567154331250748100501368753612041473388671875, 4733420323042829666698406743930280585197500897720601642504334449768066406250, 5522323710216634611148141201251994016063751047340701916255056858062744140625, 6626788452259961533377769441502392819276501256808842299506068229675292968750, 7731253194303288455607397681752791622489251466276982682757079601287841796875, 9277503833163946146728877218103349946987101759532379219308495521545410156250, 10823754472024603837850356754453908271484952052787775755859911441802978515625, 12988505366429524605420428105344689925781942463345330907031893730163574218750, 15153256260834445372990499456235471580078932873902886058203876018524169921875, 18183907513001334447588599347482565896094719448683463269844651222229003906250, 21214558765168223522186699238729660212110506023464040481485426425933837890625, 25457470518201868226624039086475592254532607228156848577782511711120605468750, 29700382271235512931061378934221524296954708432849656674079596996307373046875, 35640458725482615517273654721065829156345650119419588008895516395568847656250, 41580535179729718103485930507910134015736591805989519343711435794830322265625, 49896642215675661724183116609492160818883910167187423212453722953796386718750, 58212749251621605344880302711074187622031228528385327081196010112762451171875, 69855299101945926413856363253289025146437474234062392497435212135314941406250, 81497848952270247482832423795503862670843719939739457913674414157867431640625, 97797418742724296979398908554604635205012463927687349496409296989440917968750, 114096988533178346475965393313705407739181207915635241079144179821014404296875, 136916386239814015771158471976446489287017449498762289294973015785217285156250, 159735783946449685066351550639187570834853691081889337510801851749420166015625, 191682940735739622079621860767025085001824429298267205012962222099304199218750, 223630097525029559092892170894862599168795167514645072515122592449188232421875, 268356117030035470911470605073835119002554201017574087018147110939025878906250, 313082136535041382730049039252807638836313234520503101521171629428863525390625, 375698563842049659276058847103369166603575881424603721825405955314636230468750, 438314991149057935822068654953930694370838528328704342129640281200408935546875, 525977989378869522986482385944716833245006233994445210555568337440490722656250, 613640987608681110150896116935502972119173939660186078981496393680572509765625, 736369185130417332181075340322603566543008727592223294777795672416687011718750, 859097382652153554211254563709704160966843515524260510574094951152801513671875, 1030916859182584265053505476451644993160212218629112612688913941383361816406250, 1202736335713014975895756389193585825353580921733964714803732931613922119140625, 1443283602855617971074907667032302990424297106080757657764479517936706542968750, 1683830869998220966254058944871020155495013290427550600725226104259490966796875, 2020597043997865159504870733845224186594015948513060720870271325111389160156250, 2357363217997509352755682522819428217693018606598570841015316545963287353515625, 2828835861597011223306819027383313861231622327918285009218379855155944824218750, 3300308505196513093857955531947199504770226049237999177421443164348602294921875, 3960370206235815712629546638336639405724271259085599012905731797218322753906250, 4620431907275118331401137744726079306678316468933198848390020430088043212890625, 5544518288730141997681365293671295168013979762719838618068024516105651855468750, 6468604670185165663961592842616511029349643056506478387746028602123260498046875, 7762325604222198796753911411139813235219571667807774065295234322547912597656250, 9056046538259231929546229979663115441089500279109069742844440042972564697265625, 10867255845911078315455475975595738529307400334930883691413328051567077636718750, 12678465153562924701364721971528361617525300390752697639982216060161590576171875, 15214158184275509641637666365834033941030360468903237167978659272193908691406250, 17749851214988094581910610760139706264535420547053776695975102484226226806640625, 21299821457985713498292732912167647517442504656464532035170122981071472167968750, 24849791700983332414674855064195588770349588765875287374365143477916717529296875, 29819750041179998897609826077034706524419506519050344849238172173500061035156250, 34789708381376665380544797089873824278489424272225402324111200869083404541015625, 41747650057651998456653756507848589134187309126670482788933441042900085449218750, 48705591733927331532762715925823353989885193981115563253755681216716766357421875, 58446710080712797839315259110988024787862232777338675904506817460060119628906250, 68187828427498264145867802296152695585839271573561788555257953703403472900390625, 81825394112997916975041362755383234703007125888274146266309544444084167480468750, 95462959798497569804214923214613773820174980202986503977361135184764862060546875, 114555551758197083765057907857536528584209976243583804772833362221717834472656250, 133648143717896597725900892500459283348244972284181105568305589258670806884765625, 160377772461475917271081071000551140017893966741017326681966707110404968261718750, 187107401205055236816261249500642996687542961197853547795627824962139129638671875, 224528881446066284179513499400771596025051553437424257354753389954566955566406250, 261950361687077331542765749300900195362560145676994966913878954946994781494140625, 314340434024492797851318899161080234435072174812393960296654745936393737792968750, 366730506361908264159872049021260273507584203947792953679430536925792694091796875, 440076607634289916991846458825512328209101044737351544415316644310951232910156250, 513422708906671569823820868629764382910617885526910135151202751696109771728515625, 616107250688005883788585042355717259492741462632292162181443302035331726074218750, 718791792469340197753349216081670136074865039737674189211683852374553680419921875, 862550150963208237304019059298004163289838047685209027054020622849464416503906250, 1006308509457076276854688902514338190504811055632743864896357393324375152587890625, 1207570211348491532225626683017205828605773266759292637875628871989250183105468750, 1408831913239906787596564463520073466706735477885841410854900350654125213623046875, 1690598295887888145115877356224088160048082573463009693025880420784950256347656250, 1972364678535869502635190248928102853389429669040177975196860490915775299072265625, 2366837614243043403162228298713723424067315602848213570236232589098930358886718750, 2761310549950217303689266348499343994745201536656249165275604687282085418701171875, 3313572659940260764427119618199212793694241843987498998330725624738502502441406250, 3865834769930304225164972887899081592643282151318748831385846562194919586181640625, 4639001723916365070197967465478897911171938581582498597663015874633903503417968750, 5412168677902425915230962043058714229700595011846248363940185187072887420654296875, 6494602413482911098277154451670457075640714014215498036728222224487464904785156250, 7577036149063396281323346860282199921580833016584747709516259261902042388916015625, 9092443378876075537588016232338639905896999619901697251419511114282450866699218750, 10607850608688754793852685604395079890213166223218646793322762966662859344482421875, 12729420730426505752623222725274095868255799467862376151987315559995431213378906250, 14850990852164256711393759846153111846298432712506105510651868153328003082275390625, 17821189022597108053672511815383734215558119255007326612782241783993603698730468750, 20791387193029959395951263784614356584817805797508547714912615414659204315185546875, 24949664631635951275141516541537227901781366957010257257895138497591045178222656250, 29107942070241943154331769298460099218744928116511966800877661580522886041259765625, 34929530484290331785198123158152119062493913739814360161053193896627463249511718750, 40751118898338720416064477017844138906242899363116753521228726212732040457763671875, 48901342678006464499277372421412966687491479235740104225474471455278448549316406250, 57051566457674208582490267824981794468740059108363454929720216697824856640869140625, 68461879749209050298988321389978153362488070930036145915664260037389827969042968750, 79872193040743892015486374954974512256236082751708836901608303376954799297216796875, 95846631648892670418583649945969414707483299302050604281929964052345759156660156250, 111821070257041448821680924936964317158730515852392371662251624727736719016103515625, 134185284308449738586017109924357180590476619022870845994701949673284062819324218750, 156549498359858028350353294911750044022222722193349320327152274618831406622544921875, 187859398031829634020423953894100052826667266632019184392582729542597687947053906250, 219169297703801239690494612876450061631111811070689048458013184466363969271562890625, 263003157244561487628593535451740073957334173284826858149615821359636763125875468750, 306837016785321735566692458027030086283556535498964667841218458252909556980188046875, 368204420142386082680030949632436103540267842598757601409462149903491468376225656250, 429571823499450429793369441237842120796979149698550534977705841554073379772263265625, 515486188199340515752043329485410544956374979638260641973247009864888055726715918750, 601400552899230601710717217732978969115770809577970748968788178175702731681168571875, 721680663479076722052860661279574762938924971493564898762545813810843278017402286250, 841960774058922842395004104826170556762079133409159048556303449445983824353636000625, 1010352928870707410874004925791404668114494960090990858267564139335180589224363200750, 1178745083682491979353005746756638779466910786772822667978824829224377354095090400875, 1414494100418990375223606896107966535360292944127387201574589795069252824914108481050, 1650243117155488771094208045459294291253675101481951735170354760914128295733126561225, 1980291740586586525313049654551153149504410121778342082204425713096953954879751873470, 2310340364017684279531891263643012007755145142074732429238496665279779614026377185715, 2772408436821221135438269516371614409306174170489678915086195998335735536831652622858, 3234476509624757991344647769100216810857203198904625400933895331391691459636928060001, 3696544582428294847251026021828819212408232227319571886781594664447647382442203497144, 2310340364017684279531891263643012007755145142074732429238496665279779614026377185715}
puis 2310340364017684279531891263643012007755145142074732429238496665279779614026377185715, qui boucle.
Si on veut 1000 termes avant la boucle, on partira de 5^499.
JMF (cont.)
Méthode
(Method)
On part de n=5^k, (We start with n=5^k)
le terme suivant est (the next term is)
5^k - 0 + 5^(k-1) = 5^(k-1)*[5+1] = 6*5^(k-1), donc 6*n/5 (thus 6*n/5)
le terme suivant est (the next term is)
6*5^(k-1) - 2*5^(k-1) + 3*5^(k-1) = 7*5^(k-1), donc 7*n/5 (thus 7*n/5)
le terme suivant est (the next term is)
7*5^(k-1) - 0 + 7*5^(k-2) = 7*5^(k-2)*[5+1], donc n*(6*7)/(5*5) (thus n*(6*7)/(5*5))
... on continue ainsi, avec les puissances de 5 qui diminuent (elles sont au dénominateur) et les puissances de 7 qui augmentent. (... we continue like this, with the powers of 5 decreasing (they are in the denominator) and the powers of 7 increasing.)
...
On arrive à la boucle : 5*7^(k-1) , qui donne
(We arrive at the loop: 5*7^(k-1), which gives)
5*7^(k-1) - 0 + 7^(k-1) = 6*7^(k-1), qui donne (which gives)
6*7^(k-1) - 2*7^(k-1) + 3*7^(k-1) = 7^k , qui donne (which gives)
7^k - 0 + 7^(k-1) = 8*7^(k-1) qui donne (which gives)
8*7^(k-1) - 4*7^(k-1) + 7^(k-1) = 5*7^(k-1) , et ça recommence. (and it begins again.)
____________________
Many thanks, Jean-Marc, very convincing method!
(Dall-e creation)
Idea B
(coming soon, lavoro in progresso)
1 1 1 2 1 3 1 4 3 5 1 6 1 7 5 8 1 9 1 10 7 11 1
12 5 13 9 14 1
15 1 16 11 17 7 18 1
19 13 20
On peut remarquer le "dernier élément" des cycles records sont de la forme x = 8*7^k. Pour ces nombres, ev = 4*7^k et od = 7^k, d'où le successeur x' = (8-4+1)*7^k = 5*7^k => ev=0, od = x'/5 => x'' = (5+1)*7^k = 2*3*7^k => ev = 2*7^k, od = 3*7^k => x"' = (6-2+3)*7^k = 7^(k+1) => ev=0, od =7^k => x"" = 8*7^k = x, et la boucle est bouclée.
RépondreSupprimer*leS dernierS élémentS... On a (392, [2744], 19208, [134456], 941192, 6588344, 46118408, 322828856, 2259801992) = 8 * 7^(k = 2, 3, 4, ..., 10). Il est fort probable qu'un prochain record se produit avec 8*7^11 = 15818613944, etc.
Supprimer