| World!Of Numbers |  |
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 Squares containing at most three distinct digits Part of the [On-Line Encyclopedia of Integer Sequences] | |||
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References LINK 2 : http://www.asahi-net.or.jp/~KC2H-MSM/mathland/overview.htm ( Hisanori Mishima's website ) LINK 3 : http://djm.cc/rpa-output/arithmetic/digits/squares/three.digits.s LINK 4 : https://erich-friedman.github.io/mathmagic/0999.html LINK 5 : https://www.worldofnumbers.com/crump.htm LINK 6 : http://blue.kakiko.com/mmrmmr/htm/eqtn06.html LINK 7 : http://www.immortaltheory.com/NumberTheory/TriDigitalSquares.htm LINK 8 : http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series002 LINK 9 : http://mathforum.org/rec_puzzles_archive/arithmetic/part2 LINK 10a : https://bbs.emath.ac.cn/forum.php?mod=redirect&goto=findpost&ptid=19296&pid=99359 LINK 10b : https://github.com/emathgroup/selectedTopics/tree/master/data | ||||
| TRIPLES | ROOT | SQUARE | SOURCE | PATTERNS & SPORADIC RECORDS (mostly L⩾17) |
| 0 1 2 | A058411 | A058412 | LINK 2 LINK 1 LINK 10 |
Four infinite patterns 1(0n)1 2 = 1(0n)2(0n)1 [n⩾0] 1(0n)11 2 = 1(0n)22(0n-1)121 [n⩾1] 11(0n)1 2 = 121(0n-1)22(0n)1 [n⩾1] 101(0n)10401(0n)101 2 = [17] 10959977245460011 2 = [33] 120121101221001210210111000120121 [18] 110000500908955011 2 = [35] 12100110200221012201210212022010121 [20] 10099510939154979751 2 = [39] 102000121210111101102120011101220022001 [24] 471287714788971663493899 2 = [48] 222112110111011100020110111110102200012010222201 [29] 10000009999995510010001000001 2 = [57] 100000200000010200110220220200010211120011021020002000001 |
| 0 1 3 | LINK 10 |
|||
| 0 1 4 | A058413 | A058414 | LINK 2 LINK 10 |
1(0n)2 2 = 1(0n)4(0n)4 [n⩾0] 2(0n)1 2 = 4(0n)4(0n)1 [n⩾0] 102(0n)201 2 = [17] 10677612092787462 2 = [33] 114011400004041044011001104401444 [18] 105423154192999799 2 = [35] 11114041440001011101141014414040401 [19] 3743127183788194652 2 = [38] 14011001114014141144414101441441401104 [22] 3180252254777039538502 2 = [44] 10114004404014444004140001011411401140404004 |
| 0 1 5 | A058415 | A058416 | LINK 2 LINK 10 |
[17] 23452400954944999 2 = [33] 550015110551505101015151115110001 |
| 0 1 6 | A058417 | A058418 | LINK 2 LINK 1 LINK 10 |
1(0n)3(0n)1 2 = 1(0n)6(0n-1)11(0n)6(0n)1 [n⩾1] 1(0n)8(0n)1 2 = [17] 12649351807945204 2 = [33] 160006101161166601100660666601616 [26] 77470059130002034719700749 2 = [52] 6001610061606011616611060006010661000616100111161001 |
| 0 1 7 | A058419 | A058420 | LINK 2 LINK 10 |
[16] 8427200114569499 2 = [32] 71017701771000177071770101111001 |
| 0 1 8 | A058421 | A058422 | LINK 2 LINK 10 |
9(0n)1 2 = 81(0n-1)18(0n)1 [n⩾1] 1(0n+1)9 2 = 1(0n)18(0n)81 [n⩾0] 1(0n)4(0n)1 2 = 1(0n)8(0n-1)18(0n)8(0n)1 [n⩾1] 1(0n+1)9(0n)1 2 = [18] 283160940117244651 2 = [35] 80180118008081811188010188188111801 [18] 331680389653656009 2 = [36] 110011880880801080101881180101808081 [28] 2981276371121751737986262751 2 = [55] 8888008801008880800188080010010188818118011188010088001 |
| 0 1 9 | A058473 | A058474 | LINK 2 LINK 10 |
[20] 43694278824566964251 2 = [40] 1909190001999001011109190090109911991001 [27] 100990098979999970099500001 2 = [53] 10199000091990191001091091099001091999900190199000001 |
| 0 2 3 | - | - | - | combination impossible |
| 0 2 4 | A058423 | A058424 | LINK 2 LINK 1 LINK 10 |
2(0n)6(0n)2 2 = [16] 1562062816343832 2 = [31] 2440040242204024220420044444224 [24] 205524700326856587391168 2 = [47] 42240402444444204240022400420200244000244404224 [27] 634050802727999251005000002 2 = [54] 402020420440020222440222024020044204422022004020000004 |
| 0 2 5 | A058425 | A058426 | LINK 2 LINK 1 LINK 10 |
5(0n)5 2 = 25(0n-1)5(0n)25 [n⩾1] 5(0n+1)505 2 = 25(0n)505(0n-1)255025 [n⩾1] 505(0n+2)5 2 = 255025(0n-1)505(0n+2)25 [n⩾1] 15(0n+2)85(0n)15 2 = [17] 14899671139156245 2 = [33] 222000200055005555555250522500025 [17] 50000050049995505 2 = [34] 2500005005002055502050050520205025 [20] 23500043724538482665 2 = [39] 552252055055220520520522052500505502225 [20] 23558545158870178045 2 = [39] 555005050002525502502022522050000022025 [21] 150167406766664999985 2 = [41] 22550250055025025225250200022000050000225 [22] 5000000500000500254955 2 = [44] 25000005000005252550050255205255020002052025 [22] 5000005004997549999955 2 = [44] 25000050050000550000025505552050220500002025 [23] 44949994999999949999995 2 = [46] 2020502050500020505000050500052500000500000025 [24] 447241797269721007814765 2 = [48] 200025225225050225520202505522022522200552005225 [25] 5000000500499975499999955 2 = [50] 25000005005000005500225025500555205002205000002025 [27] 500000000000500000500254955 2 = [54] 250000000000500000500255205000500255205255020002052025 [29] 50000005004999999999955050005 2 = [58] 2500000500500025050020505000050050550050002020502050500025 [29] 50050000004999999999955050005 2 = [58] 2505002500500500000020500505500050500050002020502050500025 [30] 141422082876067219949805050005 2 = [59] 20000205525005225202000505202222520222202205205000550500025 |
| 0 2 6 | - | - | - | combination impossible |
| 0 2 7 | - | - | - | combination impossible |
| 0 2 8 | - | - | - | combination impossible |
| 0 2 9 | A058427 | A058428 | LINK 2 LINK 10 |
[17] 96385117673990673 2 = [34] 9290090909029029202290909290992929 [18] 151400161921673747 2 = [35] 22922009029909029220029029909020009 [21] 149067065510873088673 2 = [41] 22220990020022929092929022220290920900929 |
| 0 3 4 | A058429 | A058430 | LINK 2 LINK 10 |
[19] 5773251280200207952 2 = [38] 33330430344333340030340440344044034304 [23] 20832739723817975138362 2 = [45] 434003044400343443044430000434430333044043044 |
| 0 3 5 | - | - | - | combination impossible |
| 0 3 6 | A058431 | A058432 | LINK 2 LINK 10 |
[17] 25107103902348156 2 = [33] 630366666363306003336330636600336 |
| 0 3 7 | - | - | - | combination impossible |
| 0 3 8 | - | - | - | combination impossible |
| 0 3 9 | A058433 | A058434 | LINK 2 LINK 10 |
[9] 969071253 2 = [18] 939099093390990009 |
| 0 4 5 | A058435 | A058436 | LINK 2 LINK 1 LINK 10 |
[20] 21082112192576920612 2 = [39] 444455454500400455000455440400550454544 [20] 63639964217112858462 2 = [40] 4050045045555405040055544045540445005444 [22] 6674983479713230005962 2 = [44] 44555404454444540454555540045000554555545444 [26] 21214250022106461574572502 2 = [51] 450044404000444005405445044455050544404404054540004 [28] 2108436491907081488939581538 2 = [55] 4445504440405440505004450045555054500055550554550445444 |
| 0 4 6 | A058437 | A058438 | LINK 2 LINK 10 |
8(0n)254(0n+2)8 2 = [18] 635613598504629262 2 = [36] 404004646604004046006460044066664644 [19] 2457776365832743262 2 = [37] 6040664664446006640606666600406400644 [19] 2542962918459579238 2 = [37] 6466660404660460644444606044000660644 |
| 0 4 7 | A058439 | A058440 | LINK 2 LINK 10 |
[16] 2010988315424552 2 = [31] 4044074004774077447400004400704 |
| 0 4 8 | A058441 | A058442 | LINK 2 LINK 1 LINK 10 |
2(0n)2 2 = 4(0n)8(0n)4 [n⩾0] 2(0n)2(0n-1)2 2 = [17] 21919954490920022 2 = [33] 480484404884004840840444000480484 [18] 220001001817910022 2 = [35] 48400440800884048804840848088040484 [20] 20199021878309959502 2 = [39] 408000484840444404408480044404880088004 [24] 942575429577943326987798 2 = [48] 888448440444044400080440444440408800048040888804 [29] 20000019999991020020002000002 2 = [57] 400000800000040800440880880800040844480044084080008000004 |
| 0 4 9 | A058443 | A058444 | LINK 2 LINK 10 |
Two infinite patterns (9n)7 2 = (9n)4(0n)9 [n⩾1] 2(0n)1(0n)2 2 = 4(0n)4(0n)9(0n)4(0n)4 [n⩾0] [17] 99704560597822753 2 = [34] 9940999404004909449099404004499009 [22] 3015775265159011230138 2 = [43] 9094900449944904494440090444449999999499044 |
| 0 5 6 | A058445 | A058446 | LINK 2 LINK10 |
[16] 2236081408416666 2 = [31] 5000060065066660656065066555556 |
| 0 5 7 | - | - | - | combination impossible |
| 0 5 8 | - | - | - | combination impossible |
| 0 5 9 | A058447 | A058448 | LINK 2 LINK 10 |
[11] 70778174997 2 = [22] 5009550055905955950009 [12] 771395165003 2 = [24] 595050500590005595990009 |
| 0 6 7 | A058449 | A058450 | LINK 2 LINK 10 |
[20] 26012881552428213576 2 = [39] 676670006660660066767076066770670707776 |
| 0 6 8 | - | - | - | combination impossible |
| 0 6 9 | A058451 | A058452 | LINK 2 LINK 10 |
[17] 30000101109940614 2 = [33] 900006066606660060090966606696996 [29] 24691314243454114014126412353 2 = [57] 609660999069000006699096996606066090996096009966990996609 |
| 0 7 8 | - | - | - | combination impossible |
| 0 7 9 | A058453 | A058454 | LINK 2 LINK 10 |
[16] 8819172285373497 2 = [32] 77777799799099990007000790009009 |
| 0 8 9 | A058455 | A058456 | LINK 2 LINK 10 |
[15] 301345331969667 2 = [29] 90809009099908808089808090889 [27] 299831600904572582192518303 2 = [53] 89898988900998890088080098089989880890999988989999809 |
| 1 2 3 | A030175 | A030174 | LINK 2 LINK 10 |
[20] 56843832676142723489 2 = [40] 3231221313313311221231322223122312333121 [21] 557963558954625926861 2 = [42] 311323333121312322332133323111223321313321 |
| 1 2 4 | A053880 | A053881 | LINK 2 LINK 10 |
One infinite pattern (3n)8 2 = (1n)4(2n-1)44 [n⩾1] [16] 4705573731461671 2 = [32] 22142424142222114221142142112241 [21] 379766258564954821662 2 = [42] 144222411144424121442444112111142224442244 [28] 1114110597927523626433041668 2 = [55] 1241242424414424212214142144114212224412421422224222224 |
| 1 2 5 | A031153 | A031153 | LINK 2 LINK 10 |
(3n)5 2 = (1n)(2n+1)5 [n⩾0] 123(3n)5 2 = 152(1n)5(2n+2)5 [n⩾0] (3n)504485 2 = [18] 159722338802442489 2 = [35] 25511225512522225551152512152515121 [21] 124588166420914599285 2 = [41] 15522211212125512112255222511252122511225 [21] 150051695267169681235 2 = [41] 22515511252551552115222525221211511125225 [23] 34816980372445012123335 2 = [46] 1212222122255221215111111512151155125251522225 [25] 1102340925268369741032335 2 = [49] 1215155515521525522215211555521512552151515552225 [28] 4638162516046117503822620335 2 = [56] 21512551525255251211125255525551555121211151225555512225 |
| 1 2 6 | A053882 | A053883 | LINK 2 LINK 10 |
[20] 47130268582155593596 2 = [40] 2221262216626122626662262611111116211216 [26] 51595698572871617009432954 2 = [52] 2662116111222626216162621266666216222222216621166116 |
| 1 2 7 | A053884 | A053885 | LINK 2 LINK 10 |
[18] 130834904430015239 2 = [35] 17117772217211221211117217772227121 |
| 1 2 8 | A053886 | A053887 | LINK 2 LINK 10 |
One infinite pattern (3n)59 2 = (1n)28(2n-1)881 [n⩾1] [17] 28616100692061141 2 = [33] 818881218818182112888822882221881 [17] 33350084135098989 2 = [34] 1112228111818181281181888828822121 [19] 1490909832561385391 2 = [37] 2222812128828218222281282181228222881 [20] 34377642169166984891 2 = [40] 1181822281111288118221882821111822281881 |
| 1 2 9 | A053888 | A053889 | LINK 2 LINK 10 |
[17] 99961098929489127 2 = [34] 9992221299191112291912929211222129 [18] 459556524411439511 2 = [36] 211192199129121999192121999211919121 [27] 145303131149776986249167839 2 = [53] 21112999921929291129929211912291229929191119991929921 |
| 1 3 4 | A053890 | A053891 | LINK 2 LINK 10 |
[20] 21079405433537116521 2 = [39] 444341333431434111311131411343131143441 [30] 177324875114669443080086908188 2 = [59] 31444111334433114334141133143444444313434111431113141443344 |
| 1 3 5 | - | - | - | impossible since all 3 digits odd |
| 1 3 6 | A053892 | A053893 | LINK 2 LINK 10 |
[17] 18266544874814631 2 = [33] 333666661663616663311166611666161 [29] 11537606482136410218512760694 2 = [57] 133116363336636111166666336113333136136363666113311361636 |
| 1 3 7 | - | - | - | impossible since all 3 digits odd |
| 1 3 8 | A053894 | A053895 | LINK 2 LINK 10 |
[24] 286074095527510693610891 2 = [47] 81838388131883313833383381811133318888113813881 |
| 1 3 9 | - | - | - | impossible since all 3 digits odd |
| 1 4 5 | A053896 | A053897 | LINK 2 LINK 10 |
[14] 73560479506012 2 = [28] 5411144145154411455544144144 |
| 1 4 6 | A027677 | A027676 | LINK 2 LINK 9 LINK 10 |
[19] 1290017214657004546 2 = [37] 1664144414111416144464164661464666116 [21] 375720828696801774892 2 = [42] 141166141116611464114116414644641441611664 [21] 802565644925350914229 2 = [42] 644111614414444441664646611416446114664441 [22] 4051067284576580130696 2 = [44] 16411146144166666464466464146441416441444416 [23] 10789398111648380852704 2 = [45] 116411111611641646616166611414441166144111616 [30] 105567345643273687982611367608 2 = [59] 11144464466166416111166414141141166144146161144464111641664 |
| 1 4 7 | A053898 | A053899 | LINK 2 LINK 10 |
[19] 1071033028175028538 2 = [37] 1147111747441771474111741117114417444 [22] 2177492084289725902412 2 = [43] 4741471777144414774147714111744447747417744 |
| 1 4 8 | A053900 | A053901 | LINK 2 LINK 10 |
[18] 284800189808379191 2 = [35] 81111148114888814414411114441814481 [19] 2117642891144336478 2 = [37] 4484411414414144114118188814881444484 [19] 6435147099182306059 2 = [38] 41411118188114448414441181181148111481 [25] 1346900557360669225841779 2 = [49] 1814141111418481411488184148411114184811141884841 [28] 2209483759119790145920022988 2 = [55] 4881818481814118844811411488844844184118148818448448144 |
| 1 4 9 | A027675 | A006716 * (* seq by Neil Sloane) |
LINK 2 LINK 9 LINK 4 LINK 10 |
[18] 648070211589107021 2 = [36] 419994999149149944149149944191494441 |
| 1 5 6 | A053902 | A053903 | LINK 2 LINK 10 |
One infinite pattern (3n)4 2 = (1n+1)(5n)6 [n⩾0] [17] 12472031176057954 2 = [33] 155551561656561551165551166666116 [19] 2482270463831785216 2 = [37] 6161666655611666116165616661556166656 [20] 12516036335682176169 2 = [39] 156651165556116515661615565555551516561 [21] 258100003219026842869 2 = [41] 66615611661661666651111615111161616151161 [23] 74240565428619726479296 2 = [46] 5511661555161166511651661611666516615516655616 |
| 1 5 7 | - | - | - | impossible since all 3 digits odd |
| 1 5 8 | A053904 | A053905 | LINK 2 LINK 10 |
[19] 3399291958357679641 2 = [38] 11555185818155188818511188881585888881 [22] 2412237158970509643109 2 = [43] 5818888111118115811551585855811558551185881 |
| 1 5 9 | - | - | - | impossible since all 3 digits odd |
| 1 6 7 | A053906 | A053907 | LINK 2 LINK 10 |
[10] 1292931424 2 = [19] 1671671667166667776 |
| 1 6 8 | A053908 | A053909 | LINK 2 LINK 10 |
[17] 12723468913060546 2 = [33] 161886661181618111866616661818116 [18] 126931550889393381 2 = [35] 16111618611186661611161686166611161 [21] 431495267861269619604 2 = [42] 186188166186668818681818186188818861116816 [25] 1080794204132598414568541 2 = [49] 1168116111686616811868118886618818666111186868681 |
| 1 6 9 | A053910 | A053911 | LINK 2 LINK 10 |
[19] 3019927482025216937 2 = [37] 9119961996691166966116961161911661969 [19] 4082512947923373236 2 = [38] 16666911969961991191619191116961111696 [21] 411900436901564744737 2 = [42] 169661969919699919691616999616691969199169 [23] 34149670012924966713187 2 = [46] 1166199961991666696199696161116991961919696969 [30] 248216864092061020657513399437 2 = [59] 61611611619696691696969619616966119999999161919199911916969 |
| 1 7 8 | A053912 | A053913 | LINK 2 LINK 10 |
[9] 105769141 2 = [17] 11187111187877881 [9] 279067891 2 = [17] 77878887787187881 |
| 1 7 9 | - | - | - | impossible since all 3 digits odd |
| 1 8 9 | A053914 | A053915 | LINK 2 LINK 10 |
[16] 2969848344609859 2 = [31] 8819999189981919818818919999881 |
| 2 3 4 | A053916 | A053917 | LINK 2 LINK 10 |
[15] 205483392086668 2 = [29] 42223424423443333243223342224 |
| 2 3 5 | A053918 | A053919 | LINK 2 LINK 10 |
[18] 159789024443333515 2 = [35] 25532532332552235533223325522255225 [27] 576387476638096486959455635 2 = [54] 332222523225232223533222222253253255253352335533253225 |
| 2 3 6 | A058457 | A058458 | LINK 2 LINK 10 |
[16] 2514602599284156 2 = [31] 6323226232326633633323632632336 [27] 251462176552105392823457806 2 = [53] 63233226236322223226263332266636366232262662262333636 |
| 2 3 7 | - | - | - | combination impossible |
| 2 3 8 | - | - | - | combination impossible |
| 2 3 9 | A053920 | A053921 | LINK 2 LINK 10 |
[19] 1814641285211195673 2 = [37] 3292922993992939999922923222293922929 [20] 14940646884386874573 2 = [39] 223222929323939222223332232999233932329 |
| 2 4 5 | A031154 | A031152 | LINK 2 LINK 10 |
Three infinite patterns (6n)5 2 = (4n)(2n+1)5 [n⩾0] (6n)515 2 = (4n)24(2n-1)45225 [n⩾1] 2(3n)5 2 = 5(4n-1)5(2n+1)5 [n⩾1] [22] 1566985170463509665838 2 = [43] 2455442524452554445254444455245254424242244 [22] 5022405325580985229335 2 = [44] 25224555254424242244454444254455442544542225 [22] 5042276720337304556485 2 = [44] 25424554524455524225542255422225542555555225 [24] 156602211177063382566485 2 = [47] 24524252545545555425452252442555424225445255225 [24] 473545618135383472428338 2 = [48] 224245452455222424254455242452522555442545442244 [27] 212706944938912242495946332 2 = [53] 45244244425245444452555552444225545225555452224254224 |
| 2 4 6 | A053922 | A053923 | LINK 2 LINK 10 |
One infinite pattern (6n)8 2 = (4n)6(2n)4 [n⩾0] [17] 14988743331128338 2 = [33] 224662426646444226244244226642244 [18] 211335426908131668 2 = [35] 44662662666442262666444262424462224 [19] 4964540927663570432 2 = [38] 24646666622446664464662466646224666624 [22] 8015264452445860907338 2 = [44] 64244464242642246466444266646244264622246244 [25] 1562826497869300353470568 2 = [49] 2442426662442422262266222264624646466242442242624 [26] 47144739098275455895604568 2 = [52] 2222626424644462446266642442624422642664422222466624 [27] 163176169897520398456349838 2 = [53] 26626462422424442244464226222466224666246222642626244 [30] 162062538622046218465618335432 2 = [59] 26264266424622222222622644266266266222662426642466466626624 |
| 2 4 7 | A058459 | A058460 | LINK 2 LINK 10 |
[11] 88002411582 2 = [22] 7744424444247727742724 [17] 47146022358675418 2 = [34] 2222747424244722424422447477474724 |
| 2 4 8 | A027679 | A027678 | LINK 2 LINK 9 LINK 8 LINK 10 |
[18] 669644852476481662 2 = [36] 448424228448248888288442822222282244 [20] 20597146608802018338 2 = [39] 424242448424484484244824228422488282244 [23] 53710727465081333454522 2 = [46] 2884842244828242284244242842824284482242248484 [25] 2069416058768323727702022 2 = [49] 4282482824288222284288288882288482848444822888484 [27] 149140498954591218312271662 2 = [53] 22242888428424424282282224442842448448442222888242244 [29] 20598117118436403877526792022 2 = [58] 424282428824822822284282828848288484482824448422442848484 |
| 2 4 9 | A053924 | A053925 | LINK 2 LINK 10 |
[21] 222468490448488507807 2 = [41] 49492229242429222429494944494949499949249 [21] 547259530974381470838 2 = [42] 299492994242299992492444422992424244422244 |
| 2 5 6 | A030486 | A030484 | LINK 2 LINK 10 |
[18] 237112320688458875 2 = [35] 56222252622266562625655622566265625 [18] 257395128676171075 2 = [35] 66252252266222665256252522666655625 [18] 745355990494250516 2 = [36] 555555552565665265566665252566266256 [19] 7453363177489241484 2 = [38] 55552622655552522252252226565666522256 [20] 80781591501545428925 2 = [40] 6525665525522556666225656625562226655625 [21] 228618071522411733125 2 = [41] 52266222626626566662522622656666222265625 [24] 237750344316525128700475 2 = [47] 56525226222626252566552565525652262562265225625 [24] 516291221749568609228465 2 = [48] 266556625655662226525525225552225265562566256225 [26] 81401637345465395512991484 2 = [52] 6626226562522666562566262626266252566552622656522256 [29] 14920674457351323857264196585 2 = [57] 222626526262256222655556652225555252265566656525525662225 [29] 23508012597117321085533117075 2 = [57] 552626656266226655522226225556662256522626266565656555625 |
| 2 5 7 | A030487 | A030485 | LINK 2 LINK 10 |
One infinite pattern 1(6n)5 2 = 2(7n)(2n+1)5 [n⩾0] [18] 870185357137045415 2 = [36] 757222555775727275772757755772522225 [26] 52174924557278712520943915 2 = [52] 2722222752557725255755757775775772222257522575527225 |
| 2 5 8 | A053926 | A053927 | LINK 2 LINK 10 |
[18] 159013392166264585 2 = [35] 25285258888222255288288552225222225 |
| 2 5 9 | A053928 | A053929 | LINK 2 LINK 10 |
[17] 77170285247817565 2 = [34] 5955252925229529299525595522529225 [19] 9794365502654705173 2 = [38] 95929595599592555525252922555552959929 [22] 1598601020441309256565 2 = [43] 2555525222555995255555255252529952995599225 [26] 30484348812550551609088485 2 = [51] 929295522525252225925225599299529559999252559595225 [26] 76975943837016723668817565 2 = [52] 5925295929599552922952299222959292529595925252529225 |
| 2 6 7 | A058461 | A058462 | LINK 2 LINK 10 |
[18] 150573163864701424 2 = [35] 22672277676226226672276776667627776 |
| 2 6 8 | - | - | - | combination impossible |
| 2 6 9 | A053930 | A053931 | LINK 2 LINK 7 LINK 5 LINK 1 LINK 10 |
[20] 78883604126137785577 2 = [40] 6222622999929222269696699966269229222929 [23] 47567102808870567435673 2 = [46] 2262629269629662226292666922999622262992962929 [26] 26395073915340646948470264 2 = [51] 696699926996296229992699669262629222696929692229696 |
| 2 7 8 | - | - | - | combination impossible |
| 2 7 9 | A053932 | A053933 | LINK 2 LINK 10 |
[11] 14907304327 2 = [21] 222227722297792922929 [27] 850281851974525726895170673 2 = [54] 722979227797229279772792797979272222799797729799272929 |
| 2 8 9 | A053934 | A053935 | LINK 2 LINK 10 |
[13] 5405829167667 2 = [26] 29222988989999289998222889 [26] 17320185602062360469701767 2 = [51] 299988829289888292222928892882898892999989922922289 |
| 3 4 5 | A053936 | A053937 | LINK 2 LINK 10 |
[20] 21343231796858797962 2 = [39] 455533543534444433554534333443535353444 |
| 3 4 6 | A053938 | A053939 | LINK 2 LINK 10 |
[17] 66817996351009092 2 = [34] 4464644636363464333646646666664464 [19] 6666680833328031344 2 = [38] 44444633333463334436443663666646446336 |
| 3 4 7 | A058463 | A058464 | LINK 2 LINK 10 |
[17] 21150351639576462 2 = [33] 447337374477734734334374744437444 [20] 20913496712251033188 2 = [39] 437374344733334774447744773373477443344 [22] 1864878916830083039312 2 = [43] 3477773374437343773773333743737447337433344 [27] 271006150065722262703289312 2 = [53] 73444333373444774773333473377377773344743344373433344 |
| 3 4 8 | A053940 | A053941 | LINK 2 LINK 10 |
[19] 6952948212333013522 2 = [38] 48343488843384848488834343333834844484 [25] 5906300402396058566810062 2 = [50] 34884384443343843348888488484833488344838384443844 |
| 3 4 9 | A053942 | A053943 | LINK 2 LINK 10 |
[21] 185605616817891584607 2 = [41] 34449444994349999433343349994949439344449 |
| 3 5 6 | A053944 | A053945 | LINK 2 LINK 10 |
[17] 23527926717739784 2 = [33] 553563335635333565566365536366656 |
| 3 5 7 | - | - | - | impossible since all 3 digits odd |
| 3 5 8 | - | - | - | combination impossible |
| 3 5 9 | - | - | - | impossible since all 3 digits odd |
| 3 6 7 | A053946 | A053947 | LINK 2 LINK 10 |
[17] 25226323103806424 2 = [33] 636367377337637773373677663667776 [19] 1932967502917049474 2 = [37] 3736363367333373666777367633763676676 |
| 3 6 8 | A058465 | A058466 | LINK 2 LINK 10 |
[15] 183539278812156 2 = [29] 33686666866886336386333368336 |
| 3 6 9 | A053948 | A053949 | LINK 2 LINK 10 |
[23] 18430047920827535573187 2 = [45] 339666666363999366939366969366969936633336969 [26] 31047286456844613647179386 2 = [51] 963933996333366963633963999333393696336393663336996 |
| 3 7 8 | - | - | - | combination impossible |
| 3 7 9 | - | - | - | impossible since all 3 digits odd |
| 3 8 9 | A058467 | A058468 | LINK 2 LINK 10 |
[12] 199974958167 2 = [23] 39989983893893399999889 |
| 4 5 6 | A030177 | A030176 | LINK 2 LINK 1 LINK 10 |
[17] 25425667278648884 2 = [33] 646464556564556546656456554445456 [21] 237605691124298293112 2 = [41] 56456464454655444665444666454556666644544 [23] 75269219840819770294592 2 = [46] 5665455455445656566644565645546455454464446464 [24] 675754811056988742949784 2 = [48] 456644564666666555445565455644644555565545646656 [26] 25781108305591628417975738 2 = [51] 664665545464645645665646644665564654546645556644644 |
| 4 5 7 | A053950 | A053951 | LINK 2 LINK 10 |
[16] 8629863583949388 2 = [32] 74474545477575775744575745574544 [29] 27341447393189418631675507588 2 = [57] 747554745554544455555455747745774457774554454557445577744 |
| 4 5 8 | A053952 | A053953 | LINK 2 LINK 10 |
[15] 767175898056538 2 = [30] 588558858558855585845444545444 [30] 293061185503724726684202222622 2 = [59] 85884858448848555885485448555548484845848584584884848554884 |
| 4 5 9 | A053954 | A053955 | LINK 2 LINK 10 |
[24] 703957001491895099962643 2 = [48] 495555459949459999994555545994544549499995545449 |
| 4 6 7 | A053956 | A053957 | LINK 2 LINK 10 |
[18] 253863101778786762 2 = [35] 64446474444746646444646744666444644 [24] 815277035409723028858892 2 = [48] 664676644466466777446466766667744446667647467664 |
| 4 6 8 | A053958 | A053959 | LINK 2 LINK 10 |
[20] 26204321529981784378 2 = [39] 686666466846666884868486448488884846884 |
| 4 6 9 | A053960 | A053961 | LINK 2 LINK 10 |
[21] 999997323321167445187 2 = [42] 999994646649499499946646996649944649464969 [23] 22236561446627600040614 2 = [45] 494464664969644944649644494946666694449496996 [25] 9718263579193026119075264 2 = [50] 94444646994669646646646969664664969996646496669696 |
| 4 7 8 | A053962 | A053963 | LINK 2 LINK 10 |
[19] 9208064263934568938 2 = [38] 84788447488748874848488744487874447844 [20] 22019193553462506122 2 = [39] 484844884744844787448744774844887478884 |
| 4 7 9 | A053964 | A053965 | LINK 2 LINK 10 |
[10] 8819171038 2 = [20] 77777777797497997444 [27] 865996535661545126193725357 2 = [54] 749949999777797799497449749797947997449449477944777449 |
| 4 8 9 | A053966 | A053967 | LINK 2 LINK 10 |
One infinite pattern (6n)7 2 = (4n+1)(8n)9 [n⩾0] [10] 6670081667 2 = [20] 44489989444449498889 [26] 30732718558321504090886022 2 = [51] 944499989984998988844994898844848999494444994984484 |
| 5 6 7 | A053968 | A053969 | LINK 2 LINK 10 |
[18] 815816631091556424 2 = [36] 665556775565576667756557666775667776 [21] 881854102200458483334 2 = [42] 777666657567776675657756675676567555755556 [22] 2562353735836753390526 2 = [43] 6565656667556566576676575665776756666556676 |
| 5 6 8 | LINK 2 LINK 10 |
Only this one rather trivial solution is known [3] 816 2 = [6] 665856 | ||
| 5 6 9 | A053970 | A053971 | LINK 2 LINK 10 |
[18] 834023722663550236 2 = [36] 695595569965566559565695699695655696 |
| 5 7 8 | - | - | - | combination impossible |
| 5 7 9 | - | - | - | impossible since all 3 digits odd |
| 5 8 9 | A058469 | A058470 | LINK 2 LINK 10 |
[11] 92496431583 2 = [22] 8555589855588599885889 |
| 6 7 8 | LINK 2 LINK 10 |
( No rootsolutions under 1025 | ||
| 6 7 9 | A053972 | A053973 | LINK 2 LINK 10 |
[16] 9831977256725526 2 = [32] 96667776776767999797799699976676 |
| 6 8 9 | A053974 | A053975 | LINK 2 LINK 10 |
[22] 2588184048685235767383 2 = [43] 6698696669868698868988986669668968886668689 |
| 7 8 9 | A058471 | A058472 | LINK 2 LINK 10 |
[16] 9949370777987917 2 = [32] 98989978877879888789778997998889 |
Farideh Firoozbakht (email) (1962–2019 †)
A058426 (squares with digits 0,2 & 5)
[ di 9/12/2008 20:37 ]
I found the following interesting infinite pattern for numbers m such that
the digits of m2 are 0, 2 & 5 which isn't in the mentioned in the table
Examples :
I hope that the pattern and examples are clear.
Best wishes,
Farideh
Farideh Firoozbakht (email) (1962–2019 †)
A058426 (squares with digits 0,2 & 5)
[ di 16/12/2008 17:22 ]
I also found the following similar patterns a(n, k) for [ 1 < k < n+1 ].
The formula for b(n, k) = a(n, k)2 is more intricate than of b(n).
I wrote the formula of b(n, 2) after the following examples :
b(n, 2) for [ n > 2 ], includes 2n substrings and I separated them with "." and put them
between parentheses as follows.
Examples :
b(3, 2) = a(3, 2)2 =
b(4, 2) = a(4, 2)2 =
Hope that the formula for b(n,2) and the examples are clear.
Best wishes,
Farideh
Farideh Firoozbakht (email) (1962–2019 †)
A058426 (squares with digits 0,2 & 5)
[ vr 19/12/2008 16:22 ]
I also found a very nice formula for b(n, k) :
b(n, k), [ 1 < k < n ] :
and
b(n, n), [ n > 2 ] :
Dear Farideh,
What intricate and beautiful patterns you discovered there! Thanks a lot !
Farideh Firoozbakht (email) (1962–2019 †)
A058426 (squares with digits 0,2 & 5)
[ di 24/02/2009 9:24 ]
Since the following six numbers are solutions of infinite patterns so they
must not be in the table of sporadic solutions.
1. 4495
2. 4949995
3. 4949994995
4. 4494999499499995
5. 4494999499999995
6. 4494999499999994995
The number 44949994999999949999995 isn't a solution of my infinite pattern.
But are you sure that the last number couldn't be made part of that
infinite pattern as well. It does look so similar to the others, perhaps
an adaptation 'somehow' of your pattern could include this one as well ?
Farideh Firoozbakht (email) (1962–2019 †)
A058426 (squares with digits 0,2 & 5) - New patterns
[ di 26/02/2009 4:29 ]
1.
For each natural number n, [ n > 1 ] there exist n numbers m of the following
form where m2 has only three distinct digits 0, 2 & 5.
Where k is in the set A(n) = {2^3–2, 2^4–2, ..., 2^n–2, 2^(n+1)–4, 2^(n+1)+2}.
Note that A(n) has n elements and A(2) has only the two last terms.
Since for [ n > 2 ] we have min(A(n)) = 6 and only for [ n = 2 ] min(A(n)) = 4,
I considered the two solutions related to A(2) = {4,10} namely 4494999499499995 &
4494999499499999999995 as two sporadic solutions.
But as you expected we can include these two solutions, specially the first one
which is correspondent to 4 ( the smallest term of A(2) ), in these infinite patterns.
Suppose that the i-th term (in increasing order) of A(n) = A(n,i) now we define :
Examples :
c(2,1) = 44.(92^1–1).4.(92^2–1).4.99.4.(9A(2,1)).5
c(2,2) = 44.(92^1–1).4.(92^2–1).4.99.4.(9A(2,2)).5
c(3,2) = 44.(92^1–1).4.(92^2–1).4.(92^3–1).4.99.4.(9A(3,2)).5
A(2,1) = 2^(2+2)–4 = 4 ; A(2,2) = 2^(2+2)+2 = 10 ; A(3,2) = 2^(3+1)–4 = 12
Hence,
c(2,1) = 4494999499499995
c(2,2) = 4494999499499999999995
c(3,2) = 44949994999999949949999999999995
cc(2,1) = 20205020500505205550255005000025
cc(2,2) = 20205020500505250500205050005005000000000025
cc(3,2) = 2020502050500020500505500500002055502550000000500500000000000025
2.
For each natural number n, there exists a number d(n),
where dd(n) = d(n)2 has only three distinct digits 0, 2 & 5.
Note that we have d(n+1) = 5.(03*2^n–1).d(n) and number of digits of d(n) equals to
8 + (3*2^0 + 3*2^1 + ... + 3*2^(n-1)) = 3*2^n + 5.
So for the sequence {d(n)} we obtain the following recursion relation.
d(1) = 50049995505, d(n+1) = d(n) + 5*10^(3*2^(n+1)+4).
Examples :
d(2) = 5.(03*2^1–1).5.(03*2^0–1).49995505 = 50000050049995505
dd(2) = 2500005005002055502050050520205025
d(3) = 5.(03*2^2–1).5.(03*2^1–1).5.(03*2^0–1).49995505
d(3) = 50000000000050000050049995505
dd(3) = 2500000000005000005005002050505005002055502050050520205025
d(4) = d(3) + 5*10^(3*2^4+4) = 50000000000050000050049995505 + 5*10^52
d(4) = 50000000000000000000000050000000000050000050049995505.
Anne Zahn (email)
Reporting an error within a pattern of 0 1 8 .
[ di 12/6/2021 20:37 ]
Hello,
I found a mistake in the table on this webpage. One of the infinite patterns is wrong.
For triple 0 1 8 there is the pattern 1(0n)4(0n)1 2 = 1(0n)818(0n)8(0n)1 [n ⩾ 1] which is not right.
It should be 1(0n)8(0n-1)18(0n)8(0n)1 .
Zhao Hui Du (email)
Update for squares containing at most three distinct digits
[ vr 1-2/3/2024 7:00 ]
Hi Patrick,
I have searched for the problem of squares containing at most three distinct digits
and found all results with length of base less than 27 digits.
The results are available at a Chinese BBS (see LINK 10).
I think we should thank the development of the IC industry for the achievement.
C++ code is used with 'openmp' to run in a machine with 40 cores to process one pattern
(square with given 3 digits) no more than 27 digits for its bases (so squares are no more than 54 digits)
The code first enumerates all integers so that the last 16 digits of its squares are in the pattern.
Next it enumerates squares whose first 16 digits are in the pattern and find square root of it.
Some digits of the two lists of integers should be overlapped for target integers so that we could
search in the first list for those integers with overlapped digits and verify the result.
The corresponding C++ code is in the BBS too.
https://bbs.emath.ac.cn/forum.php?mod=viewthread&tid=19296&page=9#pid99352
Zhao Hui Du (email)
Second Update
[ wo 6/3/2024 23:52 ]
All results no more than 30 digits are available now such as
[30] 177324875114669443080086908188^2 = [59] 31444111334433114334141133143444444313434111431113141443344
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Patrick De Geest - Belgium
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