n^2个数码对应的正整数

王守恩

A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, ...}

a(1)=1,  1^2个数码={1},  最后一个数是1。
a(2)=4,  2^2个数码={1, 2, 3, 4},  最后一个数是4。
a(3)=9,  3^2个数码={1, 2, 3, 4, 5, 6, 7, 8, 9},  最后一个数是9。
a(4)=13, 4^2个数码={1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1},  最后一个数是13(进1)。
a(5)=17, 5^2个数码={1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7},  最后一个数是17。
a(6)=23, 6^2个数码={1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2},  最后一个数是23(进1)。
a(7)=29, 7^2个数码={1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9},  最后一个数是29。
a(8)=37, 8^2个数码={1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3},  最后一个数是37(进1)。
a(9)=45, 9^2个数码={1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3,7,3,8,3,9,4,0,4,1,4,2,4,3,4,4,4,5}, 最后一个数是45。

得到一串数——1, 4, 9, 13, 17, 23, 29, 37, 45, 55, 65, 77, 89, 102, 111, 122, 133, 144, 157, 170, 183, ......——什么规律？

王守恩

{1, 4, 9, 13, 17, 23, 29, 37, 45, 55, 65, 77, 89, 102, 111, 122, 133, 144, 157, 170, 183, 198, 213, 228, 245, 262, 279, 298, 317, 336, 357, 378, 399, 422, 445, 468, 493, 518, 543, 570, 597, 624, 653, 682, ——需要这么复杂吗？？！！

RecurrenceTable[{-42441298436574738792877941474981516750698197856735821975273112901648861347858097144107828320635291605731737575530225557357312000 +
       12869945060932756129456574838265415070730879762973503496915022825615031228314328731566908589283877740567601023538847631858090240 n +
       1064691820961470094712853751779211566375273723540214843392716271419358877708038340057873380610943504314569774439379153448080576 n^2 -
       1419023478925235655168513111170571250522971380632881882350429035163512965748060251739281820237760453980191586380094069537677408 n^3 +
       401881925404369984601842899925478033647727659548842585874151907957188716984911819352573062258478922005676942341279210595376272 n^4 -
       64789984293204783435949862392822341516301901135707009753772256919594370483958426499393144076336380621628508566444727707978640 n^5 +
       6717125921296872359754983337672952966422332766163851601466366552040821999407093390347065678165336227240155433220810395188008 n^6 -
       458267244309175863097853229280684499918872473627193113508262867412275237978086889494131201916060195249081779635291728040694 n^7 +
       20343088296211565285238346997073987329006625108658652769682364435459412724639244618752762156458909568096089119941462285416 n^8 -
       570143822665927267172097235043031706118962258434243891021480300096194387946841834249351902403709948093557716691364675550 n^9 +
       9624056827867499939828583740362174818072364252599028440883714310048805714667055684873188078451173681094449042141445816 n^10 -
       88893691214594331834004012519935699378516777290111158076567673638295365202908330965198125383650787102942738815420098 n^11 +
       344444471272777498278213698628073010665594917949962825257984447060936648790741531427427998125095870251456738347112 n^12 -
       434914716817906013234512256741438361024528799133396193455510995700484999001600380372847784283058003242589050 n^13 +
       (-2994148761696040630923648274463605612920988571738924142885381167234253696378143176251403818571498179339622567437591629782656000 +
       2953316872706127202185129174993781462354636799150276055154321909418890745864196373643597802453533495601114845224143421758491520 n -
       1070743044701388112661916911157883801169820871470082253691339756939864620213906161495543819591904524627998657139609438877250432 n^2 +
       217169723906300250884715113323188033489070524992206508199726458893192103336546548323690476404549378950439461248147589255971832 n^3 -
       28487961556174851147483665934254208261053280972236178151874921573813249182872830708034186540311136818824484719335705247006256 n^4 +
       2544512499053897649680604483825856579954466386312226695425261579865995387555043417055809677055789135582153035687229812805046 n^5 -
       155056949546803014597839678216605056028659336826310784940338933719427114959412473797088505812243744717181573630086757935526 n^6 +
       6227592754475523298354733400793813595933854230309626300658932734435330483857073567865773090583902363577711188873687625486 n^7 -
       151345636573983818875008823316204984358062606733937308530448372474187599991262744025237834867337517354585976206273479518 n^8 +
       1679599170977257404168438185162865702660523811399973561575799528695049334176193108968913855015959945753219314622868818 n^9 +
       10945944420419496049178020482775747317414026557635312057437477426220879475787718932272022541054828198915035084053358 n^10 -
       575549415668476923694295557178751039723621608990977326454480543812387277080640235691478650021965561763961730725118 n^11 +
       6621328873702871197228319695173323169860503529791938290533986137404488571208744783830051858517499329155898127174 n^12 -
       26867302668241330662038210258388897376074881001701467754719091964887986282560637008847830280200660908903690384 n^13) a[n] +
       (2141491126333677650832822574340600825440544610568188294608795506457746723671824558272192276026311827864962922800715545393792000 -
       1305089682058757706949860765802883164271116891987162310567880901229030450124286172658965484736256548871313343871662569569485440 n +
       242297197744930144301294615317168251615946836457029560861407082747188856467646016989912612721713463100383722694182915357251104 n^2 -
       4982644983865277783194378056682095776278411983045332844901558962711830466486236906599743635028498574683835904411264020712016 n^3 -
       4552741684783070158905225011014118594275000695154263169958671892247366713572317095759471607308908431823267447608607017023484 n^4 +
       777855644865516290286507436585204826322879071922264280839114715033231193352379525964871237146639824256704770592111072954436 n^5 -
       62241566934558653712775040219846002452422187076092074771505191051783557065918008647338377813580983327790184268795149499513 n^6 +
       2692373025249908623806521899813783264842111910817512066781077176142502953865000212276593197082734372650056219483818172647 n^7 -
       53246865586870785507890378852355286572980372316059749293951261666833975278426585679025119009068994511739046431370702877 n^8 -
       290892736005367870515766133849345079651544898241833027828194412757497615877151647751522311575001605506441527794116317 n^9 +
       39008315023969162126017702416957450977676258687598297455005539527046947491548400716749366706732114310239608034758709 n^10 -
       846031404507072041813695222585349982009619821861715582527752790475995895193980997607602131337640044403441311427131 n^11 +
       8213021459748806950864846783528394099742097937258853289879023724808826586331214472446018597596855716318012096061 n^12 -
       31199779704932527442463620933174458505416890643262739316345972452130722618151465179984979498825508201529018179 n^13) a[1 + n] +
       (-3944307077091529538640623938498645128617773487858268694645648996449181585186956542591606017140004494800152884542749497874944000 +
       1051956594393679777393563306235732148374223912018547488789298907042944215006929082633587189669693282538931933756651142916782080 n -
       53132028448812597615808321841980275512450767188643736134640221732409782393439503411265955242354440079614217903285323537523328 n^2 -
       15177762721723915175692659934882809653494271852472808442691043661210353014531281802802930540664084076102886625723692017644736 n^3 +
       3182206666346392772676774422311538108494936723670578379633679857814403000881084752861656715064061715922640375767131857434064 n^4 -
       304230148502712965316311771762360486702915839645470411190213903066305931682766966661924555342379976916432354500465498612000 n^5 +
       17956130020081873321943107168571840740824127492808363155369355129113374305470649242565221821590878691255033623762469321876 n^6 -
       706715307982582939757213886180210446024118039918535373766207748538676571484683841219394733366348530910721327792698588548 n^7 +
       19098729243019765612425893608835959408669177346762137871021198582894884915397584667970227386060762854101173108314147492 n^8 -
       363003876635268716762644852246518979018045215936963079195545436828450868075170821361159405454084800429027455486747060 n^9 +
       5019863192569772669120254787959851459581515323310615410322935901050359098282735724157377720923911476577145271144252 n^10 -
       51545886861233949391882341501837317794648105239634585518560718024038778567506595455841665583190147764744634663116 n^11 +
       363183643375956460249572570391968985938424431664247921729492736010480214678014309944192729289915066668904298044 n^12 -
       1272284962750273687421654938812022076375050980099806601200845385241457886860293602317997763081630252772885020 n^13) a[2 + n] +
       (-2100571904170850252699024142715043038719041017126506694771273663142961375003738301258135526835272704544065218340442256759168000 +
       1290008757232392434725649190490448223400980529108403568217337075255226186776616443068284936914187793575730219601405749252556160 n -
       241112628827190070767497392001277529804987746718432532036408828291794305554815397132719488519284843797027828494665539295106336 n^2 +
       5488497390136649314836118781245186993826232580832314830193208947274862066920169070341098642996343046567927373402669610843072 n^3 +
       4367783894116275731211190714662880186822260471101152159366122811042546890345954565971334451085971483519636086427167311366180 n^4 -
       746321675102918106770063102471372883754877586430835012421375156872870857408676905727403440427434621379397090759962995935856 n^5 +
       58914409849454986016819332723021743936436327812815193566199807935900177284006855312293907338462033052602899078146661044607 n^6 -
       2463942767460424766108798534238012998329160497085083128859563804855765838342934940452555916441130161382977767354594075939 n^7 +
       43081236087861552993386292096821578313202646006964987732310427384640137628976328019594755345515129563664078009953023515 n^8 +
       576100927269459803022680437313270921804325353623965524249087090234229439295108018751697501089670686903198610298677017 n^9 -
       43824938276051307838225143112243694448145940854905959405073097626786536058197712360496535385718354007612419074070371 n^10 +
       890529522167682843314028582752407502446331596710635260106560127296144862665760047249258857503749070744260768971767 n^11 -
       8385434799757137875772770176305922650409981344512384252732993728343122892471480538046015873241472636862230663995 n^12 +
       31199779704932527442463620933174458505416890643262739316345972452130722618151465179984979498825508201529018179 n^13) a[3 + n] +
       (6979375060950397567698070644587808528260265653038874437368552006998220630233185975857066584902541797460673156440614416292224000 -
       4020354391926172251802904056541948551598997074047582286294164642435639224218795185867865539945295533435629903251051384992202880 n +
       1125059642067940783811522456315754798493230728397323018650978233127668953520176284764002899036687584010968769242412138476918528 n^2 -
       201486108778304964177380712663742132618028432541946717771743765247418718721581334357146190855917450402452483153432491648196040 n^3 +
       25120797099161663947112857215591431745105604024512488761648692634794026358765383425384392669024138154598212982387133683914888 n^4 -
       2208748380788586500847848377949664150683549061175327015817308118639329119928573830156417324994203955788413001354616237174466 n^5 +
       133773662441617473579940863551208956771849350070225540579664195474430360872030671219478813515533915750739254815675800158744 n^6 -
       5292447188703456500899796149037832883396784776658661988971211614609916796850324454822341076575949621399911408951764940230 n^7 +
       122081277831954820748078842951835316689615703078080409097786339609098877426414901697837243817722889552409834676541652664 n^8 -
       1031387103077896754898879029452420881489698140080877985959361414389866642683065916607579260047206063927434776631561058 n^9 -
       20782430865071414430505715966021842247465224048253589417827971427010827140719766300176568940964979372864991394509272 n^10 +
       671593420190321674586511258847645877954981489079531589551848598656575023119925880788977041771264735869525822932870 n^11 -
       7156925857087158582385815658342820706466811368709717175117448876949265092027025159374241863452031316369020993152 n^12 +
       28139587630991604349459865197200919452449931981801274355919937350129444169420930611165828043282291161676575404 n^13) a[4 + n] == 0,
a[1] == 1, a[2] == 4, a[3] == 9, a[4] == 13}, a, {n, 50}]

王守恩

楼上的题目——电脑还可以来一个复杂的——楼下的这串数——电脑连复杂的也来不了了。

A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, ...}

a(1)=1,  1^1个数码={1},  最后一个数是1。
a(2)=4,  2^2个数码={1, 2, 3, 4},  最后一个数是4。
a(3)=18,  3^3个数码={1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8}, 最后一个数是18。
a(4)=122, 4^4个数码={1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6,...1, 1, 6, 1, 1, 7, 1, 1, 8, 1, 1, 9, 1, 2, 0, 1, 2, 1, 1},——最后一个数是122(进1)。
a(5)=1058, 5^5个数码={1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, ...1, 0, 4, 9, 1, 0, 5, 0, 1, 0, 5, 1, 1, 0, 5, 2, 1, 0, 5, 3, 1, 0, 5, 4, 1, 0, 5, 5, 1, 0, 5, 6, 1, 0, 5, 7, 1, 0, 5, 8}} 最后一个数是1058。
a(6)=11553, 6^6个数码=
a(7)=155775, 7^7个数码=
a(8)=2555475, 8^8个数码

得到一串数——{1, 4, 18, 122, 1058, 11553, 155775, 2555475, 49816449, 1111111111, 26947525611, 752267629947, 22427586978811, 748207862444608, 26412058911292381, 1029360799201087511, 41917568649872393764,
1838978144836711303933, 86501337685714117401525, 4238748444444444444444444, 220507338129521986388601315, 12156516843976919603715787403, 699719303698634104848871467389, 42026465248793601111256018248590, 2644969208268342186617695428187551, 174089741425507456164105149986425413, 11961515772477602133141072150796342208, 852734210881140665991893317037838895483, 62897494250544445016413154288832371648648}

王守恩

这些数字串可都是OEIS没有的。

a(n)=  n^2——{1, 4, 9, 13, 17, 23, 29, 37, 45, 55, 65, 77, 89, 102, 111, 122, 133, 144, 157, 170, 183, 198, 213, 228, 245, 262, 279, 298, 317, 336, 357, 378, 399, 422, 445, 468, 493, 518, 543, 570, 597, 624, 653, 682,——1楼
a(n)=10^n——{1, 10, 55, 370, 2777, 22222, 185185, 1587301, 13888888, 123456790, 1111111111, 10101010100, 91919191919, 842592592592, 7777777777777, 72222222222222, 674074074074074, 6319444444444444,——10^n
a(n)= n^n——{1, 4, 18, 122, 1058, 11553, 155775, 2555475, 49816449, 1111111111, 26947525611, 752267629947, 22427586978811, 748207862444608, 26412058911292381, 1029360799201087511, 41917568649872393764,——3楼
a(n)=n(n+1)/2——{1, 3, 6, 10, 12, 15, 19, 23, 27, 32, 38, 44, 50, 57, 65, 73, 81, 90, 100, 106, 113, 121, 128, 136, 145, 153, 162, 172, 181, 191, 202, 212, 223, 235, 246, 258, 271, 283, 296, 310, 323, 337, 352, 366, 381, 397, 412, 428, 445}
Table[Module[{t = k = 0}, While[t < n (n + 1)/2, k++; t += IntegerLength[k];]; k], {n, 50}]
{1},
{1, 2, 3},
{1, 2, 3, 4, 5, 6},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 4, 0, 4, 1, 4, 2, 4, 3, 4},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 4, 0, 4, 1, 4, 2, 4, 3, 4, 4, ...,4, 8, 4, 9, 5, 0},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 4, 0, 4, 1, 4, 2, ...,2, 5, 3, 5, 4, 5, 5, 5, 6, 5, 7},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 4, 0, ...,7, 5, 8, 5, 9, 6, 0, 6, 1, 6, 2, 6, 3, 6, 4, 6},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, ...,6, 3, 6, 4, 6, 5, 6, 6, 6, 7, 6, 8, 6, 9, 7, 0, 7, 1, 7, 2, 7},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, ...,6, 9, 7, 0, 7, 1, 7, 2, 7, 3, 7, 4, 7, 5, 7, 6, 7, 7, 7, 8, 7, 9, 8, 0, 8, 1},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, ...,5, 7, 6, 7, 7, 7, 8, 7, 9, 8, 0, 8, 1, 8, 2, 8, 3, 8, 4, 8, 5, 8, 6, 8, 7, 8, 8, 8, 9, 9, 0},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, ...,2, 8, 3, 8, 4, 8, 5, 8, 6, 8, 7, 8, 8, 8, 9, 9, 0, 9, 1, 9, 2, 9, 3, 9, 4, 9, 5, 9, 6, 9, 7, 9, 8, 9, 9, 1},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, ...,9, 0, 9, 1, 9, 2, 9, 3, 9, 4, 9, 5, 9, 6, 9, 7, 9, 8, 9, 9, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 3, 1, 0, 4, 1, 0, 5, 1, 0, 6},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, ...,9, 8, 9, 9, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 3, 1, 0, 4, 1, 0, 5, 1, 0, 6, 1, 0, 7, 1, 0, 8, 1, 0, 9, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 3},
Table[First[RealDigits[ChampernowneNumber[], 10, n (n + 1)/2]], {n, 21}]

王守恩

A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, ...}

a(1)=1,  1^1个数码={1},  最后一个数是1。
a(2)=4,  2^2个数码={1, 2, 3, 4},  最后一个数是4。
a(3)=18,  3^3个数码={1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8}, 最后一个数是18。
a(4)=122, 4^4个数码={1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6,...1, 1, 6, 1, 1, 7, 1, 1, 8, 1, 1, 9, 1, 2, 0, 1, 2, 1, 1},——最后一个数是122(进1)。
a(5)=1058, 5^5个数码={1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, ...1, 0, 4, 9, 1, 0, 5, 0, 1, 0, 5, 1, 1, 0, 5, 2, 1, 0, 5, 3, 1, 0, 5, 4, 1, 0, 5, 5, 1, 0, 5, 6, 1, 0, 5, 7, 1, 0, 5, 8}} 最后一个数是1058。
a(6)=11553, 6^6个数码=
a(7)=155775, 7^7个数码=
a(8)=2555475, 8^8个数码

得到一串数——{1, 4, 18, 122, 1058, 11553, 155775, 2555475, 49816449, 1111111111, 26947525611, 752267629947, 22427586978811, 748207862444608, 26412058911292381, 1029360799201087511, 41917568649872393764,
1838978144836711303933, 86501337685714117401525, 4238748444444444444444444, 220507338129521986388601315, 12156516843976919603715787403, 699719303698634104848871467389, 42026465248793601111256018248590, 2644969208268342186617695428187551, 174089741425507456164105149986425413, 11961515772477602133141072150796342208, 852734210881140665991893317037838895483, 62897494250544445016413154288832371648648}

这串数若用4#的代码——Table[Module[{t = k = 0}, While[t < n^n, k++; t += IntegerLength[k];]; k], {n, 10}]——连“10”也出不来。

这串数改用这个代码——Table[Module[{t = k = p = 0}, While[True, t = p + 9 (k + 1) 10^k; If[t > n^n, Break[]]; p = t;  k++]; 10^k - 1 + Ceiling[(n^n - p)/(k + 1)]], {n, 29}]——效果很好——还可以改吗？谢谢！！！

王守恩

n^9个数码对应的正整数——这些按钮我还是不熟悉——基本功太差——继续讨教。

a(1)=1,  
a(2)=207,  
a(3)=5198,  
a(4)=54650,
a(5)=344039,
a(6)=1598400,
a(7)=5923531,
a(8)=18166104,

这串数若用4#的代码——Table[Module[{t = k = 0}, While[t < n^9, k++; t += IntegerLength[k];];k], {n, 10}]——连“10”也出不来。
{1, 207, 5198, 54650, 344039, 1598400, 5923531, 18166104, 49816449, 123456790}

Table[Module[{t = k = p = 1}, While[True, t = p + 9 k*10^(k - 1); If[t > n^9, Break[]]; p = t; k++]; 10^(k - 1) + Floor[(n^9 - p)/k]], {n, 49}]——效果很好——还可以改吗？谢谢！！！
{1, 207, 5198, 54650, 344039, 1598400, 5923531, 18166104, 49816449, 123456790, 274339866, 585654606, 1171561048, 2177215789, 3955447048, 6983058784, 11790817055, 19042763770, 30345346262, 47555555555, 73217377971,
109865027408, 159355314381, 229409887611,327150698061,461717899173,644725716341,890797255376,1201404391306,1599547008546,2119287174752,2791960246149,3655584270235,4756007990582,6148211521768,7897774444578,
10076632207584, 12594800883854, 15702819447847, 19518222222222, 24178074678933, 29841606782898, 36693123074853, 44945210758615, 54842268120659, 66664376698583, 80731541729562, 97408326550414, 115968313934770}