Earlier in the web book, I mentioned a hierarchy of functions that I invented based on reptends in reciprocals of powers. I originally intended for it to be a fast iteration hierarchy like FGH, but I eventually changed it to a polyadic function.
We will define r(a) as simply (the smallest integer of the form 10^n - 1 (where n is a positive integer) divisible by a) divided by a if a is coprime to 10. I will add a definition for when a is not coprime to 10 eventually.
If the number is not known to be prime or composite and has no known prime factors (as in 10^10^100 + 37), then the value of the function is unknown and we'll have to live with it.
If the first argument is of the form 2a5b, then the result is just zero, as the reciprocal of any such number terminates.
Below are the first few values of r(n):
r(1) = 9 (since every integer is divisible by 1, it follows from the definition that 9 is the smallest number of the form 10^n - 1 divisible by 1, and 9/1 = 9).
r(2) = 0 (because 1/2 terminates)
r(3) = (10^1 - 1)/3 = 3
r(6) = ((10^1 - 1)/3)*(5 mod 3) = 3*2 = 6 (Normally, we would simply multiply by 5, but since 5 is greater than 3 we use the modulo operator)
r(7) = (10^6 - 1)/7 = 142,857
r(9) = 1
r(11) = (10^2 - 1)/11 = 9
r(13) = (10^6 - 1)/13 = 76,923
Here are the first values of r(n):
9, 0, 3, 0, 0, 6, 142857, 0, 1, 9, 3, 76923, 714285, 6, 0, 588235294117647, 5, 52631578947368421, 0,
47619, 45, 434782608695652173913, 6, 0, 384615, 37, 571428, 344827586206896551724137931, 3, 32258064516129, 0, 3, 2941176470588235, 285714, 7, 27, 263157894736842105, 25641, 0,
2439, 238095, 23255813953488372093, 27, 2, 2173913043478260869565, ...
Joyce already gave names to some numbers expressible with this function:
r(7) = (10^6 - 1)/7 = 142,857 (integral-megaseptile)
r(17) = (10^16 - 1)/17 = 588,235,294,117,647 (integral-dekapetaseptemdecile)
r(19) = (10^18 - 1)/19 = 52,631,578,947,368,421 (integral-exaundevigintile)
r(29) = (10^28 - 1)/29 = 344,827,586,206,896,551,724,137,931 (integral-myriayottaundetrigintile)
We can also extend Joyce's system, and name:
r(13) = (10^6 - 1)/13 = 76,923 (integral-megatredecile)
r(21) = (10^6 - 1)/21 = r(7)/3 = 47,619 (integral-megaunvigintile)
r(31) = (10^15 - 1)/31 = 32,258,064,516,129 (integral-petauntrigintile)
r(39) = (10^6 - 1)/39 = r(13)/3 = 25,641 (integral-megaundequadragintile)
r(41) = 99999/41 = 2,439 (integral-dekamyriaunquadragintile)
r(51) = (10^16 - 1)/51 = r(17)/3 = 196,078,431,372,549 (integral-dekapetaunquinquagintile)
r(53) = (10^13 - 1)/53 = 188,679,245,283 (integral-dekateratrequinquagintile)
Next, we will define r(a, b) as r(ab).
r(3, 2) = (10^1 - 1)/9 = 1
r(3, 3) = (103 - 1)/27 = 37
r(3, 4) = (109 - 1)/81 = 12,345,679
r(3, 5) = (1027 - 1)/243 = 4,115,226,337,448,559,670,781,893
r(3, 6) = (1081 - 1)/729 = 1,371,742,112,482,853,223,593,964,334,705,075,445,816,186,556,927,297,668,038,408,779,149,519,890,260,631
r(3, 7) = (10243 - 1)/2187 = 457,247,370,827,617,741,197,988,111,568,358,481,938,728,852,309,099,222,679,469,593,049,839,963,420,210,333,790,580,704,160,951,074,531,321,444,901,691,815,272,062,185,642,432,556,012,802,926,383,173,296,753,543,667,123,913,037,494,284,407,864,654,778,235,024,148,605,395,518,975,765,889,346,135,259,716,506,630,086,877 (240 digits)
r(3, 8) = (10729 - 1)/6561 = 152415790275872580399329370522786160646242950769699740893156531016613321140070111263526901386983691510440481633897271757354061880810852004267642127724432251181222374638012498094802621551592745008382868465172991921963115378753238835543362292333485749123609205913732662703856119493979576284103033074226489864349946654473403444596860234720317024843773814967230605090687395214144185337600975461057765584514555707971345831428135954884926078341716201798506325255296448712086572168876695625666819082456942539247065996037189452827312909617436366407559823197683279987806736777930193568053650358177107148300563938424020728547477518670934308794391098917847889041304679164761469288218259411675049535131839658588629782045419905502210028959 (726 digits)
r(3, 9) = (102187 - 1)/19683 = 50805263425290860133109790174262053548747650256566580297718843672204440380023370421175633795661230503480160544632423919118020626936950668089214042574810750393740791546004166031600873850530915002794289488390997307321038459584412945181120764111161916374536401971244220901285373164659858761367677691408829954783315551491134481532286744906772341614591271655743535030229131738048061779200325153685921861504851902657115277142711984961642026113905400599502108418432149570695524056292231875222273027485647513082355332012396484275770969872478788802519941065894426662602245592643397856017883452725702382766854646141340242849159172890311436264797032972615963013768226388253823096072753137225016511710613219529543260681806635167403342986333384138596758624193466443123507595386882080983589899913631052177005537773713356703754508967128994563836813493877965757252451353960270284001422547375908144083727074124879337499364934207183864248336127622821724330640654371792917746278514454097444495249707869735304577554234618706497993192094701011024742163288116648884824467814865620078240105674947924604989076868363562465071381395112533658487019255194838185235990448610476045318294975359447238733932835441751765482904028857389625565208555606360818980846415688665345729817609104303205812122135853274399227759995935578925976731189351216786059035716100187979474673576182492506223644769598130366305949296347101559721587156429406086470558349845043946552862876594015139968500736676319666717471930091957526799776456840928720215414316923233246964385510338871107046690037087842300462327897170146827211299090585784687293603617334755880709241477417060407458212670832698267540517197581669460956155057663973987705126251079611847787430777828583041203068637910887567952039831326525428034344358075496621449982218157801148198953411573439008281257938322410201696895798404714728445866991820352588528171518569323781943809378651628308692780572067266168775085098816237362190722958898541888939694152314179749021998679063150942437636539145455469186607732561093329268912259310064522684550119392369049433521312808006909515825839556978102931463699639282629680434893054920489762739419803891683178377279886196209927348473301834070009653 (2,183 digits)
There are a number of interesting patterns in the values of r(3, n).
r(3,4), the reptend in 1/81, contains all the nonzero digits in order except eight.
In r(3, 5) each block of three digits is the previous block plus 111, with each block of three being 111n + 4.
In r(3, 6), all integers of the form 100 + 37n appear in order, up to 951 (988 becomes 989 because the next term in the sequence would be 1025, which adds the 1 to 988).
In r(3, 7), each block of 9 digits (starting by considering the first 8 digits as the first block) is 37,037,037 more than the previous one, and each is of the form 8,687,700 + 37,037,037n. Additionally, each block of three consists of three digits from each multiple of 12,345,679:
12345679*
24691358*
37037037
49382716
61728395
74074074
86419753
98765432*
111111111
123456790*
135802469
148148148
In r(3, 8), each block of 9 digits (starting by considering the first 8 digits as the first block) is of the form 12,345,679n + 2,895,900. Each block of three digits is also a block of three digits from a certain multiple of r(3, 5).
The prime factorization of each value of r(3, n) is given below.
r(3, 1): 3 (prime)
r(3, 2): 1
r(3, 3): 37 (prime)
r(3, 4): 37 * 333667
r(3, 5): 37 * 757 * 333667 * 440334654777631
r(3, 6): 37 * 163 * 757 * 9397 * 333667 * 2462401 * 440334654777631 * 676421558270641 * 130654897808007778425046117
r(3, 7): 37 * 163 * 757 * 9397 * 333667 * 2462401 * 440334654777631 * 676421558270641 * 411361786890737698932559 * 130654897808007778425046117 * 810316718654935254370114242173495345974018609204623064367629310709846283851718998731856856949061264781563433202832207926987960406521712613
r(3, 8): 37 * 163 * 757 * 9397 * 313471 * 333667 * 2064529 * 2462401 * 386953775911 * 440334654777631 * 676421558270641 * 411361786890737698932559 * 130654897808007778425046117 * 810316718654935254370114242173495345974018609204623064367629310709846283851718998731856856949061264781563433202832207926987960406521712613 * 1331071238411335998086135185950861382071163049829915112408307256210123464739174398604774383795837375882614453767323958324349448000539213624287620258183754970659808004836778292079711281761095569691768792999666858328321421564465274233961249916692588824087815924183945723161041419839993990678786583150584157403751902905229512502548735734146578079001534082483518676082847550221421498391272218372186125985275484874724609745193816551314986555505001043826263264309520883
Red denotes a new factor.
r(3, 8) is the largest one that is fully factored. r(3, 9) has a 1,379-digit factor known to be composite, but no prime factors of this factor are known.
r(6, 1) = 6
r(6, 2) = 7
r(6, 3) = 296
r(6, 4) = 49,382,716
r(6, 5) = 8,230,452,674,897,119,341,563,786
r(6, n) when n is at least 2 is equal to r(3, n) * (5n mod 9). Similarly, r(15, n) is equal to r(3, n) * 2n+2.
r(7, 1) = 142,857
r(7, 2) = 20,408,163,265,306,122,448,979,591,836,734,693,877,551
r(7, 3) = 2,915,451,895,043,731,778,425,655,976,676,384,839,650,145,772,594,752,186,588,921,282,798,833,819,241,982,507,288,629,737,609,329,446,064,139,941,690,962,099,125,364,431,486,880,466,472,303,206,997,084,548,104,956,268,221,574,344,023,323,615,160,349,854,227,405,247,813,411,078,717,201,166,180,758,017,492,711,370,262,390,670,553,935,860,058,309,037,900,874,635,568,513,119,533,527,696,793 (292 digits)
r(7, 4) = 416493127863390254060807996668054977092877967513536026655560183256976259891711786755518533944189920866305705955851728446480633069554352353186172428154935443565181174510620574760516451478550603915035401915868388171595168679716784673052894627238650562265722615576842982090795501874219075385256143273635985006247396917950853810912119950020824656393169512703040399833402748854643898375676801332778009162848812994585589337775926697209496043315285297792586422324031653477717617659308621407746772178259058725531028738025822573927530195751770095793419408579758433985839233652644731361932528113286130778842149104539775093710953769262807163681799250312369845897542690545605997501041232819658475635152019991670137442732194918783840066638900458142440649729279466888796334860474802165764264889629321116201582673885880882965431070387338608912952936276551436901291128696376509787588504789670970428987921699291961682632236568096626405664306538942107455226988754685547688463140358184089962515618492294877134527280299875052061640982923781757600999583506872136609745939192003331945022907122032486463973344439816743023740108288213244481466055810079133694294044148271553519366930445647646813827571845064556434818825489379425239483548521449396084964598084131611828404831320283215326947105372761349437734277384423157017909204498125780924614743856726364014993752603082049146189087880049979175343606830487296959600166597251145356101624323198667221990837151187005414410662224073302790503956684714702207413577675968346522282382340691378592253227821740941274468971261974177426072469804248229904206580591420241566014160766347355268638067471886713869221157850895460224906289046230737192836318200749687630154102457309454394002498958767180341524364847980008329862557267805081216159933361099541857559350270720533111203665139525197834235735110370678883798417326114119117034568929612661391087047063723448563098708871303623490212411495210329029571012078300708038317367763431903373594335693461057892544773011245314452311536859641815910037484381507705122865472719700124947938359017076218242399
r(11, 1) = 9
r(11, 2) = 82,644,628,099,173,553,719
r(11, 3) = 75131480090157776108189331329827197595792637114951164537941397445529676934635612321562734785875281743050338091660405709992486851990984222389181066867017280240420736288504883546205860255447032306536438767843726521412471825694966190833959429 (239 digits)
r(11, 4) =
683013455365070691892630284816610887234478519226828768526739976777542517587596475650570316235229834027730346287821870090840789563554402021719827880609248002185643057168226214056416911413154839150331261525852059285567925688136056280308722081825011952735468888737108121029984290690526603374086469503449217949593606994057782938323884980534116522095485281060036882726589713817362202035380096987910661840038248753500443958745987295949730209685130797076702411037497438699542380984905402636431937709172870705552899392118024725087084215559046513216310361314117888122396011201420667987159347039136670992418550645447715319991803838535619151697288436582200669353186257769278054777679120278669489788948842292193156205177241991667235844546137558909910525237347175739362065432689023973772283313981285431322997063042141930196024861689775288573184891742367324636295335018099856567174373335154702547640188511713680759510962365958609384604876716071306604740113380233590601734854176627279557407280923434191653575575438836145072057919541014957994672495048152448603237483778430435079571067550030735605491428181135168362816747489925551533365207294583703298954989413291441841404275664230585342531247865582951984154087835530359948090977392254627416160098353937572570179632538761013591967761764906768663342667850556655966122532613892493682125537873096099993169865446349293081073697151833891127655214807731712314732600232224574824124035243494296837647701659722696537121781299091592104364455979782801721193907519978143569428317737859435830885868451608496687384741479407144320743118639437196912779181749880472645311112628918789700157093094733966259135304965507820504063930059422170616761150194658834779045147189399631172734102861826377979646199030120893381599617512464995560412540127040502697903148692029232975889625025613004576190150945973635680622908271292944471006078819752749129157844409534867836896386858821118776039887985793320128406529608633290075814493545522846800081961614643808483027115634177993306468137422307219452223208797213305102110511577078068437948227580083327641554538624410900894747626528242606379345673109760262277166860187145686770029369578580698039751383102247114268151082576326753637046649819001434328256266648452974523598114882863192404890376340413906153951232839286933952598866197664093982651458233727204425927190765658083464244245611638549279420804589850420053275049518475513967625162215695649204289324499692643945085718188648316371832525100744484666347927054162967010450105867085581585957243357694146574687521344170480158459121644696400519090226077453725838399016460624274298203674612389864080322382350932313366573321494433440338774673861075063178744621269039
r(15, 3) = 296 (same as r(6, 3))
r(15, 4) = 197,530,864
The first composite number that will have any particularly interesting results is 21, because it is the first product of two primes that are coprime to 10. The first three values of r(21, n), however, are simply r(7, n)/3^n.
r(21, 4) = 5141890467449262395812444403309320704850345277944889217969878805641682220885330700685413999310986677361798838961132449956551025550053732755384844792036240044014582401365686108154524092327785233518955579208251705822162576292799810778430797867143834102045958216998061507293771628076778708459952386094271419830214776764825355690273085802726230325841598922259758022634601837711653066366380263367629742751219913513402337503406502434685136337225744417192424966963353746638489106905044708737614471336531589204086774543528673752191730811750248096215054426910597950442459674724009029159660840904767046652372211167157717206308071225466755107182706793979874640710403586982790092605447318761215748582123703600865894354718455787454815637517289606696798144805919344306127590870059285997089689995423717483970156467726924481054704572683192702629048596006807862978902823412055676389981540613221857147999033324592119538669587260452177847707488135087746360827021662784539363742473557828271142168129534504655981818275307099408168407196589898241987649179097186871725258508543251011666949470642376376098436351108848679305433435656953635573655010000976959188815359855204364436628770933921565602809528951414276973071919621968212833130228659869087468698741779402615165491744694854510209223523120510486885608362770656259480360549359577542279194368601560049567824106210889495631964047901851594757328479388732061229631686385816609334587954607390953357911569767740807585316817581151886302517983761909903795229354024300574349165214082609612250039849651122731783567546444125647235462590175904072891439266560743723037211861312930311958494660146749553941001948776487163270448012916428854232547138280860341113013610584067338197561715540335559771905738863950720121759966269198533532838683470364714290856176181734976681526730117595034990564630992230603503684164519927396506599616414971128285025272391647513124675418164242265311264339447041099130506321954329728868115651400393868809806613499519233241293493965991536448290578514096492716512152858119816331672502712347221579485913791064422745666671808557134115929062479111069975987371517011944611555884636545472308348887551997367352080665977653344028465505627799116623217692216720399422051511458702906710681249068032352774821190758994451900185622245874918372488829242959466477445097464533810500768712624883664728173960438294743445375126619052760938086496881443431492022356939752469392896992508265588926424689301268504378319733033046930034296409417886580180069004170073169101351803003892411083859091633630020413305155773571711375404281138003198255870753441210195340418858397478416914762881721093577264617109126341390675695826327507571433713319038877833824383872974737892133421773849373460646541307377070253649456759272113985427882415248790370267532561021385122454121482304183956273363464811472586010972794257536725952663756356662090384150636823134393591147721371239349859369295715262673474529645569490078722343056648207279888523814665699991258786205336253927118844514374154801754413027493688329451206030409140224494937808834796201171322648484941973766074835073863256564908654315845763853538391925175209917678333616137309043042765103017775515345972100102323620302240321676667643625855482026521871031103295437600588232269476195618080943639738586288634879499796895326535754135365408446069281832158411361521176875890189787177153552275029437322926147027216026244208945861035268226716234490772877556162298630714568518261423995146055398727896298353052483276001254621274057620024578236434407474251983484247818552969184650428576570461896020690967241015831880749276278916706516317789398450234213110792313902129256842570739558105933227410389703878527979596978625161326813416220607668615443153829937114679583095520899213804947527007779680277250734004864228382207002226438572405530617386788426632935865200199505350137031380957522842848401643348193396784261701657231297658897270170350831186594063173266283081637794951691939058314179791342084830908931977931006113707765797172988620996395534782318067060535476473280166185899907960160632658203114957245180763159383178819524786482998339169379013888246152580457731089412333338475223800782595729145777736642654038183678611278222551303212138975015554218664034018747332644320010695132172294465783289884358883387066088718178125369573377347915734699019441487857425661118566852288912541585039155495909626133144111764131200477167435379291550331394840627104961410112041793285719427604753163548110098158689023606419136059563659174932255593091355967935171044986399699713596700963076084553246846735670836739835768018469670559077750525758300296687079971822440238378042070947804669864922537420107876862007085525064145083581429548387760243931283775793008057342362492994174238100379985705544500491050539641404558800088440516040127313207974043736920316123425938780652094549081915457036934199227688051789120788148970850622940030131478139252677639460924203392619330423023328757050817303489801060257814388037906016526035962381929340141196312236156745389009723314873946555190481332366657925452872002920593785511181040821468421079694160354996117872697075806891161604475501462867837989315151608640432741501740529923231575320982512430520205058591841876584345000282803975709709431769684442182012638766768990286968906988343334310292522148693188537697769962104267254898936142862284747610306405252955301546166463561993202420802032075112735948498825078028187843542556856453843820218941696103989592813693882692910875612527701934893382901157439544222828965297381235184928090661812722065394562965019719149942667921287940724286691244903101074140918650150914485219635851317095243237128562687357633907682498547415942945583373182984456065116900879777458980568795923509237406224772599894077056370545194646263645291827993480082887274335282109820496603781346249762187565880471614193674446346943917400671530895048873668893105239072197284053455093299602531866866172016803698047624189509515068310014860063450928368323897964325563936837017497853260729839932949748304461618358605724980846458008751497575598644597672780374432463839655287663062201448984733727202143139946832852566574626827299324869781623911847429826049845486191453149665005836045680554912819247124397756079
The two-argument r function already grows double exponentially as the second argument increases.
Since Joyce's system can only be extended as far as the SI prefixes, we will come up with a new system for naming these numbers. While r(3, 4), or 12,345,679, can still be named integral-gigaunoctogintile, we will use the name tesseratertilon (from tesseract, as it is the reptend in the reciprocal of 3 to the 4th power), dropping the "integral" part, and also dropping the SI prefix for the corresponding power of 10 so that we can extend this naming scheme indefinitely.
As such, we can now name r(3, 5) penteratertilon, r(3, 6) hexeratertilon, r(3, 7) hepteratertilon, r(3, 8) octeratertilon, r(3, 9) enneratertilon, and r(3, 10) dekeratertilon, r(3, 11) hendekeratertilon, and r(3, 12) dodekaratertilon (click here for all the digits).
We can also name some larger values of r(3, n): r(3, 20) = icosaratertilon, and r(3, 100) = centeratertilon.
I will also name r(3, 10^100) (which is approximately 10^10^(4.77121254*10^99)) googoltertilon.
The first and last digits of the values of r(3, n) which I have named which are not shown above are given below.
Dekeratertilon: 16935087808430286711......8023336551 (6,557 digits)
Hendekeratertilon: 5645029269476762237......2674445517 (19,678 digits)
Dodekeratertilon: 1881676423158920745......0891481839 (59,044 digits)
Icosaratertilon: 28679719907924413133.........9387424399 (387,420,480 digits)
Centeratertilon: 19403252174826328375.........9623521999 (approximately 5.7264*10^46 digits)
Googoltertilon: 785874274299......65521999999...(101 9s)...999999
r(7, 2) can be named squareseptilon, r(7, 3) cubeseptilon, and all further values of r(7, n) can be named with the same naming scheme above.
Some notable values of r(3, n) much further along are r(3, 27) (13113726523970925072.........03360488077 w/ 847,288,906,431 digits, or r(3^^3)), r(3, 37), and r(3, 10^100) (785874274299......65521999999...999 w/ approx. 10^(4.77121254719*10^99) digits).
For r(a, r(a, b)), simply add du- at the beginning of each name.
r(7, r(7, 2))= r(7, 20408163265306122448979591836734693877551) = (106*(7^20408163265306122448979591836734693877550) - 1)/720408163265306122448979591836734693877551
= 39902066610466603238508782089..........2723017519199529190856598393 (a number with more than 1010^40 digits)
r(11, r(11, 2)) = r(11, 82644628099173553719) = 6338658544058191981167595350520986...811709 (approximately 2.8*1086065511170101243036 digits)
Notice how this generates numbers significantly larger than 10^10^10^9 that we can still know the first digits of (and, due to certain patterns involving the middle digits of these numbers, some digits in the middle as well). For example:
r(3, r(3, 7)) = r(3, 45724737082761774.........6506630086877 (240 digits)) = 10^(3^(r(3, 7) - 2))/3^(r(3, 7)) ~ 10^10^(2*10^239) (Larger than googolduplex!)
= 23464351303565835898865793325302230376199343864979026094187108113473137693723183159006425744176634058560766435387035838855361322534034843915124419936225624785907110822091985875043950062517312275179477587008985665267979327586574880614952420819760118387402996763330543282703940808371030209135225449394725391773727715624690422065250209021875359977380891750922780786767568448385542998713396976687083618664999741767263927480649048661115313093838925038047236758615511708438767129782171233908255967681960535479837894829682234321115227925371846003373900686323981161406494731128393647793946685537890261707358164896859430696886017804985567072297299795831208213752507371639344162577916639282145986720441799093951884134283397901752115560921145299591948608068448748951655969632636811937463502697957754425440591783155732338591859179730008760082418072112676344284054602532285250802725690605683828276694324135407777222929978457176464549235684742237965631419692453314896198052065255269607335579283485298937264986190066...............
..............1919225787121657977567667231720792349412673650429598258424246185161954110473851846105668300123907173
I will name this number duhepteratertilon. This number is particularly notable for being the first r(3, r(3, n)) greater than googolduplex (of course, r(3, r(3, 4)) is already vastly larger than googolplex)).
First and last digits of numbers of the form r(n, r(n, 2))
r(7, r(7, 2)): 399020666104666.........29190856598393 (more than 1010^40 digits)
r(9, r(9, 2)): 19512262283392008266261.........91 (approximately 5.7*10^11,780,770 digits)
r(11, r(11, 2)): 633865854405819198116759535......811709
r(13, r(13, 2)): 7918767786774152002071423508931869112299050827797110741..........0227 (more than 1010^75 digits)
r(17, r(17, 2)): 46036967916605129282116384309663213680868474919893...0907503 (more than 1010^269 digits)
r(19, r(19, 2)): 21733849689352029820610735996879348854988234089799...00407423181 (over 1010^339 digits)
We can extend this function into an array notation.
(a) = r(a)
(a, b) = r(a, b)
(a, b [c]) = r(a, r(a, r(a, ... r(a, b) ...))) w/ c copies of a
(a, b, [c [d]) = (a, b, [(a, b, [(a, b, [...[(a, b, [c])]...])])]) w/ d copies of a, b
We can also nest the "base argument":
(3 [3], 3)
= r(r(r(3, 3), 3), 3)
= r(r(37, 3), 3)
= r((10^4107 - 1)/50653, 3)
~ 10^10^8208
However, the period of 1/n^3 may not always be n^2*(the period of 1/n), so nesting the first argument more than three times will result in values that
For example,
(3, 3 [3])
= r(3, r(3, r(3, 3))
= r(3, r(3, 37))
~ 10^(10^(1.06*10^50031545098999689))
(3, 3, [3 [3])
= (3, 3, [ ([3, 3 [3 [2]])])
= (3, 3, [ ([3, 3, [ ([3, 3, [3]])]]))
= (3, 3, [ (3, 3, [r(3, r(3, 37))])])
= r(3, r(3, r(3, r(3, ... r(3, r(3, r(3, 3))) ... ))))
where the number of 3s (not including the second 3 in the innermost expression) is r(3, r(3, r(3, r(3, ... r(3, r(3, r(3, 3))) ... ))))
where the number of 3s (not including the second 3 in the innermost expression) is r(3, r(3, 37))
This is roughly 10^^(10^^(10^10^10^(5*10^16))), and somewhat larger than 3^^^4.
(3, 2, [4 [7]]) = (3, 2 [(3, 2 [(3, 2 [(3, 2 [(3, 2 [(3, 2 [(3, 2 [4])])])])])])])
The innermost expression (3, 2 [4]) is equal to r(3, r(3, r(3, r(3, 2))) = r(3, r(3, r(3, 1)) = r(3, r(3, 3)) = r(3, 37) = (10^(3^35) - 1)/3^37, which is the same as (3, 3 [2]). The next layer is roughly a power tower of twice that many 10s, being equal to r(3, r(3, r(3, ... r(3, r(3, 2)) ... ))) w/ approx. 2.22*10^50,031,545,098,999,689 3s.
The final outcome of (3, 2 [4 [7]]) is roughly 10^^10^^10^^10^^10^^10^^(4.44*10^50,031,545,098,999,689), or more crudely between 10^^^7 and 10^^^8.
We can even add more layers:
(3, 3 [3 [3 [3]]])
= (3, 3 [3 [(3, 3 [3 [3 [2] ] ] ] ])
= (3, 3, [3 [(3, 3 [3 [(3, 3 [3 [3] ])] ] )] ])
= (3, 3, [3 [(3, 3 [3 [the second number above] ] ) ] ])
~> (3, 3, [3 [(3, 3 [3 [3^^^4]])] ] )
~> (3, 3 [3 [3^^^3^^^4]])
~> 3^^^3^^^3^^^4
> 3^^^3^^^3^^^3 = 3^^^^4
Next, we can define (a, b, [1, 1]) = (a, a [a [a [...[a [a]]...]]]) w/ b layers.
(3, 3, [1, 1]) = (3, 3 [3 [3 [3]]])
(a, b, [c, 1]) = (a, (a, (...(a, (a, b, [1, 1]), [1, 1])...), [1, 1]), [1, 1]) w/ c layers
(3, 3 [2, 1])
= (3, (3, 3 [1, 1]) [1, 1])
= (3, (3, 3 [3 [3 [3]]]) [1, 1])
We can apply bracketed numbers to the arguments inside the brackets as well:
(3, 3, [1 [3], 1])
= (3, 3, [(3, 3 [(3, 3 [1, 1]), 1]), 1])
(a, b [[1, 1], 1])
= (a, a [a [a [a ... [a [a [a]]] ... ]], 1]
In general, [1, 1] = [a [a [a ... [a [a]] ... ] ] ] w/ b a's.
(a, b, [1, c])
= (a, b, [a [a [... [a [a]]...]], c-1]) w/ b layers when c is at least 2
(3, 3 [1, 2])
= (3, 3 [3 [3[3]], 1])
(3, 3 [1, 3])
= (3, 3, [3 [3 [3]], 2])
(a, b [1, 1, 1])
= (a, a, [1, a [a [a [...[a [a]]...]]])
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