r(3, 18) = r(9, 9)
The decimal expansion of r(3, 18) consists of blocks of 9 digits from each multiple of r(3, 14), with the last digit incremented, in much the same way that r(3, 7) consists of blocks of 3 digits from each multiple of 12345679. Since 12345679 leaves a remainder of 2779741 upon division by 4782969, the digits immediately after the initial 2 are the initial digits of 2779741/4782969, which appear somewhere in the decimal expansion of r(3, 14). Further along in the digits of this number, we can see how values much further along relate back to the more familiar values; around 2187 digits into this number, we find some digits that appear in r(3, 9), and correspond to the remainder of r(3, 9) upon division by 19683, and around 6561 digits in we find a block of digits that appear in r(3, 8). These are underlined.
The underlined digits 98902629 correspond to the point in r(3, 12) where the final digits of r(3, 12)*91 appear, followed by three zeroes and then the first digits of r(3, 12)*91.
r(3, 18) has 43,046,713 decimal digits, and is the first value of r(3, n) to be larger than the record-size primes.
25811747917131971819900315081167532159095488622957161153136637541129116689540908612089434433603226390021927828396293206888188094770589172427584231354372174157263014553677877372148998552319725144944515311888938326129674571754515543962363849063233204478248438739645491490771413486084366591153623782659569148393697887258616309268041835546802998330839440967201917914052294740663548127419765865816146858459001119065749772464925054596170312510240004368999699445426078123606931898741163377138786276220925424519816761678807338452355316706029969416511680671591945670173370722269673249005681782617335966451686554966895413732235519428090959845956933888439751569256834013236713456319033245554547839105122806243734827354471693932532308584226685026975948089312333710879188942430971945833251994062709471207135872465433804147616983674810239579249511504281850204365417545069486503074441166171776733264099514365126930599687514203720908524277868019520258258721055922264348801645335799470326929456743316433117196339091916225421934253972819697721252940755025478273039916585309973112960476388227366054457692866109618688752416499066470384843275544985438289506676039531817327296801796148680200545614406056877389362853238255037151636035439519565523030455934404646316989187425242241124733080392142089315260763092991708035348641563456392209550899616979214540199498844781025507404178615860453369052456076993904160809626152735561696118761545417387566200712735149121140054107979818279564455353315090157763958632554407828440895907288992142075376916887841726925289178497732989026297987043220112191846415226635058013155313528087565859223310205465152876826811294433113990520000608434521902634839738690227093281068054198857820346202701736820119495538605858297804172148468895252465596882770957423472768369769932327972463015501485260889235003779059294925416296193875280561116632114931846054223528689005397440402280840650118533095961272197970923525420463758693980689286673219804851363965936246598460103642066282147509240276654547302478883609070040691626921156459538720989018213747595574378617853636543213386940926606491377382986060915327583513529662598717126703125915470103079654106781120706292846581998919525394538439086013336790765343337326694665340737825561930979855843401199155473679658692496255663958960105488896845618301823990522091360015809592352251664211801663385954788777317350399606769377651577947391419455877048361270330232844241750982870707181416004046187655294606785755205631367627539182626967362069485178931773017301622372377935850470727169001121156501353752614772008095834084810109255734277904956131527674572730199615229952383855465114546381154353454961438552105074649265645834234647305914685374319477460573852096913748926686218704349423295472635676736239935931731271961715994839911525691146396751360251367603843998039040211938816689687261222779572713822061176532147735738364627380355198508873907285786322984069125987810107792208171003573329339326707628000541809238178933794077163533805771434045141582586769178333260531298333062606815304494646900308878604507672282660300911447148578659710483200592935083513355433300276485893341588343305198812032886572501332008798326616122773000784684880205187082916515548562017327896150582784484586203699722241587486097050484080102433611868163224583612561595832377362984537454342018550288908442320406033042821336173575476541200690085340323856748836017291795839945883708798891119049720677008386100096012216844834966898201401010569681047509079985699981912933882028113386641252213173475190157018257235228465162563975804594062137999108250570609341211171719934512807865461137239956351405049204818901563050786402781087811801300989014032244432998999183029785500064246731153137334458322879252780045920596625956971521968214747671747427896101798580921206776960110646084079461269793606605044577288734979630878531052600060086135506374831920673147464846651411872024145837057161940653066492825576914699521738510840607606584276444914610595104586737538292663710927276228800588809333726281058924583619530767770002995375910539413933783971864224248604466554168228309680338047376735410604471153821707142597716353612882874658701904637779753563833842561692703867295980827694427901049910656635405774835001047143895376168398775625932370345028396265330200437592241023680087296570419640351029550220819632489803604579106295020963643458722700646841628450889699847547298924605920880968172026647769782769542681569430366394483591702864223063845237157810721776255875821787009308121543360088010213626053215786426824730996609732739251175742540555205380477437784659860877931006896230519186609152207228771527362353827393986898818869644243312077384735323071671617243764306951767850357547816734080886465454851046868613084632186296166695510004376665788576814273754117325477847920428390146397239202287001398111394154995246005174496591996196669918508104510703872453168061537395870665993609852678700222279674010736174565098956343529884915301936960799200271516873750061267410149802376611010885384536283521133029182666691642113951283562599602211539204370783807461458240015798441677151463199975466449839724403424621148521651884033422919973651677467166688749907597168925157182381234359548805380811958037665891232716914979682450403391029739782296335907004649927020251115319819855991147644233137086355801899780266913038768065774652408742378103807514423946741753247851586909746531242440303667057732715834758032118430370366911596149474686146503728149493921061051574894894110775850060939859068733971888616350386156267538034107432041365267080647353165671111421265074083368884499033297126420177534802502404564359527200947805318577252634668993977755265287479413614595897120970285079579257874510607000963235065247155784783493988104485614853477715784928452764407098768594037885280765313369887362875121454921296121744351006691336864220415559900860070413054483548493998209784924410644683275901807041495939054477833773009356766363484714924305410187017754757931762354468557805160377049650567138693586234155003609011499647350865844372004806384930250810767006181751012141229319443660089954612596650767223619915466060959930283914333709903504871163383411041020083994576755593326402517653112559052084620129628714603165967301228614163460002240614589694557945798266750935828796602442985404419331059178958395254103352288112981035445443361670012243467071768628117136055754655763701748876786947630949895373241346561823166765968332666060932053596163831180338012530875722476309196956281783021547938833973233666534347903319073039526311681466077546559495463338801371447342321639576475781073106848512598929686447223497258039958748800195748036444195392051141621474............
.........569772483044901633996956727810025555979307021111111136922859028243......
.........11366670904181322222222480339701......
.........3744524957222900000004904232104255074645781059865421831110228142838361860619095961132814532171012772636296992542384613014104166287395295709308755738006411942761241003957330713089879972765198796700708.........
.........22477782015292433333333591450812......
.........33588893126403544444444702561923......
.........447000042375146555555558136730
...1791483284199767761895525355139387065303095004869502397432573577697383991480120195708079858419671758764415787002942944506995343759426208354199872996391783502188496798887680924898115029739689374043405329551375379116823116704083247388601587356935063906751715446830691497062252688447770763099728574241720086208450374445735625510503137070791317905749687905638878071830630516807798464164346248605349315946994223116578638152511340204312219532612277509153626616789490449484203712313212221463072904231453799027134055370003933891064806332429155547320575500073771266134559032059866095517730865287302861258842714433722166924424071954542393858782208082959701184002170829948025799946785984259082384256657112422363392350165558745139057423470445312457390450508671987144180182994916409802993150421633998815173660059057950339843796955199238313903423935846614452030181604566608246679488342703526968084540309379455664256311441494257160983553453725572061832795838528818748148345866137193379052288584561669891444486819591
r(3, r(3, 3)) = r(3, 37)
2220821101791888951946155768163957564902690577021182545419157180157046081399591698938207544791917840464888080850080630513425435171085231583917272838089553058998816017160265829594865038429405759947173342182989047345718686873967881786651699179426827240608196016159029342643724889666704109287452079202378749117877955706038760834875190398685307041228739364823312852206283695083866023704265665534289528197789002245718413072445518371140347932591316336653506163723319765156331376401138434811311375620174924419120982669453614300400386301891394410007716505634781202823217292660161231087744715072562580830332780443705049178253065102096425248343618526625364890399212900451143792098119454007034360290387464617179365489062685576629575088909549560673506186995729912715000895958548709748565723900292520943321944653735457973985898620074465531536847509053816907135356105294462670427190645749230257606406520776888745516647805566776215641697589551036906075001199013531810463234202343169510491003548284753822712453315112.........
......1682933157894736842105264326747948311520501024292509559977665738258198432201339694293252714234779683995631020109234101009389718362147815831910796539702721623806096798564651626080557367797903768560962......
.........6463934227551251046040371730868411111111111111111333193221290300006305726687927506867601380168813229365653026829126815719251070281004931865590302895157599919196...............
.........75750453386623621571514828419795222222222222222224443043324014111174168377990386179787124912799243404767641379402379268303621813..............
.........86861564497734732682625939530906333333333333333335554154435125222285279489101497290898236023910354515878752490513490379414732924..............
.........97972675608845843793737050642017444444444444444446665265546236333396390600202608402009347135021465626989863601624601490525844036......
.........09083786719956954904848161753128555555555555555557776376657347444507501711313719513120458246132576738100974712735712601636955147......
.........201948978310680660159592728642396666666666666666688874877684585556186128224248306242315693572436878492120858238468237127
.........313060089421791771270703839753507777777777777777799985988795696667297239335359417353426804683547989603231969349579348238
.........424171200532902882381814950864618888888888888888911097099906807778408350446470528464537915794659100714343080460690459349........
.....36559517753887000000000000000042195600934045890086976959595115193733151120963402468362963986422983875546592242279825943351046438968832873536151531979755083268250619400094428183923701508120977504326045.........
.........9402996341262583803958948388586236555616092764523436622604216816415005159675816725222397995672414267881490180009913197098999191467641887984423911348856472926837368430272924808998835134657454001045093043632156592481591217647090004321792806530857473017244117159354914281805654576572358041056669362247495532123427168740896040086766143741133884696524416313416065359102951377317433269778871937913387133400809299949883155780969066707010893448394963895514121444373434275430117524049520702481666478118793990983660304920987331820408646388571020510679251097739933505060244130191540654474666581644258249583097347284630158382186275237174383545610906722748821959318815809973136474934022569593444143889081013692986100194237440335071831771515709574384818964713548789197424186088223970170488676970818951207331042675936584292747571867874543375186367005233792144505995613489767018634721929295009780018394921187334096983751818949626626049246078106066518787525166944896951082366691029898753528231164401399349292606197573
(50,031,545,098,999,690 digits)
Interestingly, it is possible to calculate the digits of this number beginning at any given digit using modular exponentiation. The digits after the n-th digit in 1/3^m correspond to the remainder upon division of 10^n by 3^m, so the digits after the n-th digit in r(3, m) correspond to the remainder of 10^(n+floor(log10(3^m))) upon division by 3^m (the floor(log10(3^m)) part appears because the leading zeros of the repeating part in the fraction are dropped when calculating the reptend). This works for values of r(3, n) up to n in the millions or perhaps billions range.
For example, we can find that the digits in r(3, 37) starting at the 999,984th digit are ......63513656127286816027987175278666.........
r(3, r(3, 4)) = r(3, 12345679)
139686299555081665375592849638297429356678699953108034662553101141171960237661622689300342044767205758669818082607836527776813151357230021033421079113627287368497821148830376487642392148290787184449105078428650979715249683592796096537078832142252867539559648290352812618250767209795801529180052159488766096833348970182530553621005485364952395414754853986774646753841746496958033255802317175080581463417843194840767153990322921944444028078609054656090018782505207579668005754686834677359573854747569823011619426229795776705502274143857642849413143729541499165953984292894408857403682107998957190761758358663012913213495038748271624787455777004424913469253206563621244091365369354378369940661914176685053893463936300709521892353631010212192285917572489079565287169256838711250280784672935950946605452766639023949790588133535271836704069799492836265316570634516472943959206335844414307399828676164330375899397655274042991359048259643628628355963295125255604635431212593143528731856216055611014288102198917..................................................................................................................................................................................................................(7.954331345666246199345654*10^5,890,384 digits omitted)...........................................................
..................0164226016260333530978949862723902420126020869917861581694011533536171060175916319011542824061577982751957588526918745032202214844293613392723938381207859213454809100427205850175629614489183297676245029178136499805826269687574203746596163483365337787628219372036943180398549816480403265701767330335158060243124565307079143836189362300683579737380346388158505895136800840073349078698612557661359329431486024552563952168187316813518219511848360461782822919077688945682850087822972847377878321226952913155159416890926954213653565725587657714866610161955360060643354725492887246314958856148426022893775316618191230587698361733303586030941183877052561709619880041729483591769809313199177311241586307921763459087205068508032299127638396507217339761533780584172398473384835090935628012239704980506094400551803716431123533762363549787728662729009031085417776802830414860235327760568294287758608362831014177639810184311137269399643126342032874769180628495515756146324867417665021536709294102608679321287108797
r(3, r(3, 5))
43528498543677083249963734590302138233729951383017656652654200598577305779780833503447554179239987152081995160653894710670149993984076957621723248011128527290307482897380899311330268267875830375703532714460565231720321677742928774607027802658596937955046253327476366162797195257350526496256049223388723110113540108075490313555269564982707175659983382585630047492075248452905420200709427601671714651403459146213132439887227903566104897779160566034323599381462569710632159383515796575152378760376137513147707093889856427124243140283153058656589867172340724786086315019362767564494659017004579969943436055117419090435867575288787324260458247281191010479844308021955685527488509346960921629082003583876518205501814207784675490638726266346207188024373161341538682739728533497360666901928647469335926220327162780467174794237326588485083600436183524136442053605288745693413335461693955903600503555414661340317872864142534190095115860839275399134507010057312585153565125355652001417941362770677021334050870589...............
....................................(2.552*10^196346195357885776646204 digits omitted)..............................................................
.........3785240552114411262252302174729162745490166006112230161398871909169001307301013143887605405535517955079777452656733363216530429498580487551601645671245013656231997748912997084618859450052108228717703796701448593933097195469604714914187277866157679126399019665945263618599206393514497444171106672715070552544353158030686566705059846607827725595485587263057368653640459499364314576312813219548262046684352323535936657001357903437622919222917056380622380889134464710774604576692209981531841287925389874561809428304726055377601566855702992267798544193347239882625937724878194704768152248053202880744107260008503623039074731832422558127945511793656484957018324802324441613059625543638773138008831643845457510976326545575716242356032400891754503518939330723754282749360469406766734013931549466648600702541575652381650407695183363350571325024310401826959448328991160414573923490718237778195557609549652129972896099179911386489086634913717024766820956922902025049771336531671975391249623975924109559130863813
r(3, r(3, 6)
248693130455052522529459690934203626736988897839887036238827857455756449930218553685887781974268487598671598123214887730060268346711396823550754560390844375575203747617058550196015980759005414496914832696720102048095159751073505869479144193091487993244072374813705889910741830841157864934341408312197011649906022924634601669518089533981560484615013200333257459532622983866851966064054867159504535123327968829557794303040063750936247624401055849126432640038636718146766806561941379925522935942713739198756436234208440953345188241275447668215202523672113843952775983737182546095021500451811493914027633029492136604828243492380614479668630361624272033635150892499232986352585986093996485900007527600820049599665215442334407289202034344890423492170475276664680869665065053961518243297705801482790317271172932987656029232580458824186621688705488847091711927557033605857068858415696075891450926690716922229643353163534675750329464488028482485494479169485723873977381750873970214766548301994916178107979418836
.........(approximately 10^(7.272*10^724) digits omitted).........
6270754324080832093936930181567102745712580616918330553441539770650430284569640902956978123661972500613407720330181047303174307829872360845044538624513452771660457130478919726996599147277421130713518111826437459312653794074392874444056817608252329628321272101639679513928039802730568075135373503686191752492558925545567652757870055777799293471377201213302932186414072966751828307276177160310049227134939535893051628577098078227770253391846283568437697799076636123934918981383843228349071584769845990739721582450683346274985325078038634501251965048414986020487767417173409703599467137846401095753630153051214266138058552360877883382743464856547064957331854350278261424090675561170686389982794425201720729906871877297881938397145166370886034547944184454948339252478829708635794316016476023558410668586387370916141937041193893325799197150780451593619709259143467151095186563508466899620318625042958052003422829642900279166105026640574184664576911504796819403308227962673806317437678554602946803065353597
r(3, r(3, 9))
154074726139474685853132507295263964289131117381326164068561643444915268912081121979050737960423436649993242026225701518150340345082369909689948013561326550410530223953725759722626052480322480734150318860116727653168453632452275554368967552502782887837173113961421388180314055453927878689841775214121671136508124937701977829921116943301218981129061510665440302080144722475172079255345118988257104292395338392480548770962491111294639064235313820594759939872504765572118220603231308240971905441532405664303826098633153849427799101783154465787904440313987770560514828842753284001967700275012739913065998838893157659230523649919434787603775342098983737050379810530771487534059757210436124167896893999032469134669205085905311197426899564605676973215905705021464716386311687740929453405785802187291981636476900911868412031112098218507156295813976897869481956248257779543523739881420880377885448087238195901776131533374674080784527749792488864603024285129740842174968652583321904844368846561851147275040587145
..........(approximately 10^(2.424*10^2182) digits omitted)...................
8716289737570697885564158140432540957956207120360791896376859302967927710603950767564820775381952181845894895680513639937025449604680866541257798669429791286148198897826410112677402402643218730261414730262392552532073525341027322560688546245246710416744004932167839006817342553623954605451734643122429076817881519589818751418657087348927265336887983774099476737403300017809514142193620893393565124861654731476153021967926892072987308627729550469888948953368552139414351291176332136969176308766366534782495587314566996005497290524863278461945179743813314882816199344488192294640451953733903216544447606408070936423607760914450329155566337895345281687808063816474624054101150290716465537039721788117497579113629699020623718614134503195383760413111368771528354659656675202722768272260788025071588120242116848478206545030231440865664301524189790433754797250592663541659056790646304916447739291812020581198239030982756544384608109366074267377124605164928487996969020893842793572269030070379590529652670213
r(3, r(3, 10))
66790895733016545769688416916639273673871096122321415432144652084132268531233451420805211047130677530701287330780113122165046120296658036061552377612670837285104712768616515941983831136367179073314659232992273000550641296657528907547161780205534265311171762106391985596889356489880873907564492428846544825875261483471641306512339654409312202414898857023312931227358523351588462409657738704618440024432226434221900272490542289075865016626721821904077587043391309998099424620799066316690697385475776730237580345292326989862506451848063661583277066300629943065730286742762407663171013346522025773563736785523641472436183386919766867206127414938279875002685109317901205791488642058124493406488513645137106745344722743289528740723189592917070182412529593292546850750102242064164337409392890380384904696137077304421169839051531177041405205505974761153011737557380204999225803721325493892142671489999603006675492643657777291791005681608138992343251918107192484735825790285570642539933723512034761992502202245......
......(approximately 10^(8.08*10^6555) digits omitted)............
6081537642491285631309675057010526812648736939672526444118009312057809468518523160135570240203681201495510004507018466610228122501710899907375853049412910748629690117139142857880923883182001924804760711729631094822623914948745268460385905364823280113322215177522028018271822385980337543179195777895424216773565455676119402727264012918176554366688632713821472558548421329046215600730830951204488398869970950200708372674911690629403285146641394092634517051691088179751819241255717721897097598864905100342376268681857439931381645012316659100114310043590894299820934655817390590316426052865759871002976995043805117330287965504069061976209183860819197220610910531772901578209381558012466849681282542607684108332058905317406558198369989119642601879210977026481234877336002239517927751309192253378370852093202820220750438936567553286627792800778523363870360901024532145470550930669866208203115755225827468510962562000585159581255549079379498641066018492186694973454838719526307782630667536841897199719725917
More digits of r(7, r(7, 2)) are given below.
39902066610466603238508782089342338013087164695788407266616946881888438997634654734677516809870155818393431449931449877696484768922170136237819301895139567510992640493214863414699639880758447726691524629324879507526057787016231169913471388992314482634991250308249261195184473539819882898084978680979265789188100858014223205636241761566774607074980580246542610713659956477670453572271254561162064360927123170696807865178498344446778130812238337460651746143153575633713114903422715906541198632848638652.............................................................................................................................................................................
........................................................................................................................................................................................................................................................................................................................
386333713741250142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857142857
......................................................................................................................................................................................................................................................................................................................
1428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428970449237533237460956516392321994808702300218386455501237598040247455818547775118775346596670130129612
.......................................................................................................................................................................................................................................................................................................................
243476570883107285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714
........................................................................................................................................................................................................................................................................................................................
298360807805230572769824808004708091434016069999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
........................................................................................................................................................................................................................................................................................................................
999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999960097933389533396761491217910657661986912835304211592733383053118111561002365345265322483190129844181606568550068550122303515231077829863762180698104860432489007359506785136585300360119241552273308475370675120492473942212983768830086528611007685517365008749691750738804815526460180117101915021319020734210811899141985776794363758238433225392925019419753457389286340043522329546427728745438837935639072876829303192134821501655553221869187761662539348253856846424366286885096577284093458801367151361347............................................................................................................................................................................
......................................................................................................................................................................................................................................................................................................................
70163919219476942723017519199529190856598393
The repetitions of 142857 occur around every multiple of 1/7 of the way through the decimal expansion, and are analogous to the runs of repeated digits in large values of r(3, r(3, n)).
Below are the first and last digits of r(7, r(7, 7)), which has roughly 10^10^705888 digits.
137793640248561301301189229501296067523490907762257322067404947412739844709946723679635558785648559037109061083910942614135205864386205879680400765325509209353568180745092675867080395504795413644019139046701514007714299312974750227650421062560192547361811535295091236765306612449791678338965351337191739367134722005979423745334723250485128799756930104099667076453309990188000614470733682654800869544671862819254942658918317448341741346715787066997068405339099054134286138927273387510433663697316309458167094318493924155760342943888851270175333373811349389583873587640560370007081656757770650234182041873488956428650993208034680279782465891622346160169010651317963934458533214172705671974550277345370529142383670676154612635530360137412242083716030048784603615204163119743394284705450935199645782456238750536076227916708856789574228342392140531563647337775380409847540761102199341311624875213821659364595864775353354750488912353593077303899230780570008459965548416677830328867679618143087722645262322954......
...74112002056457