1572*8634=13572648

northwolves

王守恩 发表于 2025-10-30 19:06
再放宽条件。A, B是任意数。 A × B = C。C的数码集合  = {1到a的个位数 + 1到b的个位数} 。

其中:  a 是  ...

{542136, 231, 125233416}

northwolves

{{435261,35412,15413462532},{452136,31524,14253135264},{463215,52341,24245136315},{532614,23154,12332144556},{542136,45231,24521353416},{562341,41532,23355146412},{563241,45123,25415123643},{652431,23514,15341262534}}

northwolves

1. For[m=5,m<8,m++,For[n=1,n<m,n++,s=Join[Range@m,Range@n];v=Sort[s];t=Select[Permutations[s],#[[1]]>0&&#[[m+1]]>0&&#[[1]]*#[[m+1]]>5&];t=Select[Table[r=Partition[k,UpTo[m]];{FromDigits/@r,Times@@FromDigits/@r},{k,t}],Sort[IntegerDigits@#[[2]]]==v&];If[Length@t>0,Print[{{m,n},Length@t,t}]]]]

复制代码

{{5,2},1,{{{31251,42},1312542}}}
{{6,3},8,{{{212514,633},134521362},{{312615,423},132236145},{{345321,612},211336452},{{343521,621},213326541},{{536211,432},231643152},{{542136,231},125233416},{{614223,531},326152413},{{635112,243},154332216}}}
{{6,4},36,{{{213254,6341},1352243614},{{234116,5234},1225363144},{{234524,6113},1433645212},{{244631,5123},1253244613},{{263123,5441},1431652243},{{263231,5414},1425132634},{{315242,4163},1312352446},{{311642,4253},1325413426},{{322415,4163},1342213645},{{323426,4151},1342541326},{{341321,6254},2134621534},{{351644,3221},1132645324},{{361214,4253},1536243142},{{361544,3221},1164533224},{{412361,3425},1412336425},{{413264,3251},1343521264},{{415361,3422},1421365342},{{416351,3224},1342315624},{{421352,3641},1534142632},{{425246,3131},1331445226},{{451463,3221},1454162323},{{452261,3143},1421456323},{{456221,3341},1524234361},{{441653,3221},1422564313},{{512432,4163},2133254416},{{522443,3161},1651442323},{{531623,4241},2254613143},{{531164,2324},1234425136},{{543143,6122},3325121446},{{543431,2621},1424332651},{{544322,6311},3435216142},{{562343,4112},2312354416},{{633542,2141},1356413422},{{641534,2231},1431262354},{{643151,2432},1564143232},{{643541,2213},1424156233}}}
{{6,5},54,{{{212541,63354},13465322514},{{234261,53514},12536243154},{{241431,52365},12642534315},{{261342,54153},14152453326},{{263544,51321},13525341624},{{262134,55413},14525631342},{{223413,64551},14421532563},{{315423,45162},14245133526},{{315312,42546},13415264352},{{316155,42423},13412243565},{{311325,46425},14453263125},{{321153,44625},14331452625},{{326151,44532},14524156332},{{341316,45252},15445231632},{{343221,61545},21123536445},{{361251,45432},16412355432},{{331551,46224},15325613424},{{332154,46521},15452136234},{{413532,56142},23216513544},{{411453,32625},13423654125},{{421236,51354},21632153544},{{421431,55623},23441256513},{{422415,36531},15431242365},{{435261,35412},15413462532},{{435324,51261},22315143564},{{452136,31524},14253135264},{{452235,36141},16344225135},{{452463,31215},14123632545},{{453252,34116},15463145232},{{461223,35415},16334212545},{{463215,52341},24245136315},{{465234,52131},24253113654},{{442653,35112},15542432136},{{512241,63354},32452516314},{{513432,61452},31551423264},{{513456,24231},12441552336},{{513651,22434},11523246534},{{516321,45234},23355264114},{{532614,23154},12332144556},{{534462,25131},13431564522},{{541452,63231},34236551412},{{542136,45231},24521353416},{{543321,42615},23153624415},{{545313,24612},13421243556},{{561252,41433},23254354116},{{561543,22341},12545432163},{{562341,41532},23355146412},{{563241,45123},25415123643},{{554232,64311},35643214152},{{554331,22614},12535641234},{{556413,22134},12315645342},{{611325,23445},14332514625},{{611523,43245},26445312135},{{652431,23514},15341262534}}}
{{7,2},29,{{{1521263,74},112573462},{{1734221,65},112724365},{{2161742,53},114572326},{{2162741,53},114625273},{{2173421,65},141272365},{{2127614,53},112763542},{{2743112,56},153614272},{{3725162,41},152731642},{{4213157,62},261215734},{{4216172,53},223457116},{{4256711,32},136214752},{{4267121,53},226157413},{{4312721,56},241512376},{{4523612,71},321176452},{{4531622,71},321745162},{{5147261,32},164712352},{{5276231,41},216325471},{{5223716,41},214172356},{{5226113,47},245627311},{{5227136,41},214312576},{{5672231,41},232561471},{{5714162,23},131425726},{{5761412,23},132512476},{{5762231,41},236251471},{{6252371,41},256347211},{{6273521,41},257214361},{{6754211,32},216134752},{{7461221,35},261142735},{{7462121,35},261174235}}}

王守恩

A266578——Least number N such that the product n*N has the same digits as the concatenation (n,N) (counting repetitions), or 0 if no such number exists.

0, 8714, 51, 0, 251, 21, 0, 86, 351, 0, 9209, 86073, 0, 926, 93, 0, 9635, 6012, 0, 8714, 6, 0, 9017, 651, 0, 401, 81, 0, 3701, 51, 0, 926, 40611, 0, 41, 936, 0, 3251, 6882, 0, 35, 678, 0, 9203, 3141, 0, 371, 2913, 0, 251, 3, 0, 635, 846, 0,

a(1)=0。
a(2)=8714, 2*8714=17428。
a(3)=51, 3*51=153。
a(4)=0。
a(5)=251, 5*251=1255。
a(6)=21, 6*21=126。
a(7)=0。
a(8)=86, 8*86=688。
a(9)=351, 9*351=3159。
a(10)=0。
a(11)=9209, 11*9209=101299。
a(12)=86073, 12*86073=1032876。
a(13)=0。
a(14)=926, 14*926=12964。
a(15)=93, 15*93=1395。

northwolves

王守恩 发表于 2025-11-1 05:12
A266578——Least number N such that the product n*N has the same digits as the concatenation (n,N) ( ...

1. Table[k=Mod[n,3];If[k==1,0,While[Sort@IntegerDigits[n k]!=Sort@Join[IntegerDigits@n,IntegerDigits@k],k+=3];k],{n,100}]

复制代码

王守恩

A × B = C。 A是 n 位数。  B是 n 位数。A,B数码集合 = (1到n的个位数) + (1到n的个位数) = C数码集合。

好不容易找了4个。还有吗？谢谢！！！！

a(8)。  26874138*55314672 = 1486534128752736 。
a(9)。  438976458*722693151 = 317245279646839158 。
a(10)。3294118752*4636905807 = 15274518370096392864 。
a(18)。530266101869715174*743258583239784642 = 394124831615767818242403625639557708 。

northwolves

northwolves 发表于 2025-10-30 23:38
{{5,2},1,{{{31251,42},1312542}}}
{{6,3},8,{{{212514,633},134521362},{{312615,423},132236145},{{345 ...

n=10:

1. s=Select[FromDigits/@Permutations[Range[0,9]],Union[DigitCount[#^2]]=={2}&];{Length@s,s}

复制代码

{534,{3175462089,3175804269,3204957816,3206549178,3210754689,3254196708,3260974851,3275409816,3284591706,3290581476,3406829517,3410856297,3459186720,3469857012,3475806912,3501249678,3512067849,3519876240,3549716208,3564980172,3587902614,3589612047,3590846172,3591267408,3598412670,3612980754,3619072845,3642810957,3642915807,3648129057,3672980514,3684590127,3685049127,3709526418,3719086524,3745129608,3762058914,3765420198,3794256081,3794281605,3796025184,3798562041,3841792605,3851049726,3851902674,3864520719,3869154027,3946751820,3950648217,3960754182,3962751408,3974201568,3987451206,4029365187,4035896127,4035967218,4037962185,4065179382,4068259713,4069273851,4078615293,4107932568,4128306975,4138205697,4162953087,4165093872,4183596072,4192067853,4195602873,4205397618,4219738605,4231856079,4237801695,4239081675,4239865017,4253907186,4267039158,4278069315,4297158630,4320197685,4350179268,4350179826,4361958702,4375016892,4381795062,4382519067,4390275186,4395162780,4506193728,4507291683,4510897326,4518306927,4523069781,4529718063,4531678209,4532816709,4563079812,4572160839,4612035987,4613728509,4650371928,4658792103,4679283105,4681207593,4683571902,4753806192,4759086231,4781052963,4805796213,4813697025,4861920357,4872190653,4872391056,4872561930,4872659013,4875961302,4896207513,4902635187,4935102867,4935276801,4956801237,4958610327,4968071352,4975123086,4980127563,4983720516,5014387269,5017249683,5018697342,5036184972,5037612849,5043719826,5043796821,5049627318,5061382947,5067493821,5086437912,5089237641,5097813642,5103497682,5107962348,5109628734,5109738624,5134079268,5137092648,5174098362,5193728406,5201439768,5206719348,5219043687,5230718649,5234071896,5246980731,5247089316,5273948160,5273968104,5274981630,5281643079,5286379041,5287143960,5296031874,5301742698,5302796418,5309716248,5318769402,5327940681,5346017892,5360197842,5368029147,5380716294,5382690471,5386719402,5392108476,5409162378,5420768913,5437021896,5462193708,5462890317,5467892301,5479802163,5489027316,5604192837,5614870293,5617203948,5617809342,5627809413,5634208719,5637981042,5641290387,5674830219,5692740831,5694732801,5698327014,5703146298,5724938160,5728690431,5730682419,5731208946,5734802961,5736428109,5742683019,5762098134,5762134809,5762980134,5780941236,5807246913,5829460371,5832041976,5847910326,5867194203,5871290634,5890162374,5902186473,5923460781,5924631708,5926814370,5947320168,5960483712,5972846031,5978421603,5982613047,5984017623,5984627013,6017352489,6041582973,6042573198,6042713958,6051728493,6053419782,6103894572,6129487053,6147850293,6157238094,6175439802,6198370254,6198432705,6208714953,6218057349,6239875041,6259780134,6278109435,6285374091,6297483501,6307548291,6359208147,6384720591,6387210594,6398270514,6409317825,6432015987,6459827013,6492738051,6509371824,6514273089,6521708349,6527081934,6528073491,6540328791,6579081324,6580719342,6709835124,6715408932,6745820391,6749028351,6754829031,6754918032,6784359201,6795480321,6801295347,6807324951,6820941753,6835492071,6849750231,6874013529,6874351029,6894012753,6903274518,6914382057,6927403581,6938240175,6941078253,6980217453,6980723145,6982145703,7016523849,7026459183,7026984153,7031258496,7048623591,7091234856,7091853264,7094326851,7103642895,7105348269,7109368452,7138460259,7145209863,7150982364,7153920468,7153964208,7158069342,7159084236,7184903256,7189362405,7192560843,7198430625,7204563819,7206145893,7215048936,7216938054,7238906541,7239850164,7253419086,7258601349,7259164803,7291358064,7295641830,7314208596,7315482906,7319024586,7326081549,7345810692,7360412589,7395086214,7395280461,7396548201,7396820154,7438205691,7463185290,7485631029,7492581630,7493156082,7496380512,7508243196,7510493268,7519068243,7521430968,7530268941,7549326810,7564328019,7581609432,7582634901,7590614283,7591403682,7591426830,7593021864,7598236410,7610938425,7618203459,7624918350,7629310584,7641932508,7648250319,7685134029,7802154693,7810625439,7812649053,7814062593,7829143506,7832540961,7839425610,7841032695,7841962350,7908152436,7918605234,7984326015,7985632041,8015479263,8029647513,8042316795,8047321965,8056273491,8056294713,8062914357,8069574312,8074153692,8074195632,8147362059,8190247563,8192037654,8194635207,8210736495,8214756903,8259364071,8302516749,8305169247,8320946751,8341960527,8351724609,8354061972,8370421659,8379516042,8391056724,8395276014,8413752609,8421950376,8423075169,8425307916,8456913720,8460927513,8463795102,8492760153,8502139467,8504231796,8510692734,8526047139,8534019276,8537214069,8574196023,8591674023,8603925741,8625710394,8654072139,8691054723,8694170532,8695243107,8725960314,8734621509,8741506293,8746192305,8765024193,8902153674,8913576240,8916537042,8923567041,8924731056,8935106247,8952063174,8953176042,8954031267,8962043157,8967150234,8970231456,9013827564,9017628345,9017845236,9018236574,9023586174,9025816473,9026473581,9032817645,9043186752,9046128537,9052867431,9053271648,9056172348,9057124386,9063254187,9064815273,9076148352,9085173246,9107536248,9120348657,9142058763,9142853067,9148632750,9150327648,9150476283,9157864230,9160378524,9160487253,9178206534,9180376425,9182760534,9183740625,9187620345,9208374615,9231654870,9240168537,9245601387,9260458371,9265084731,9270583416,9270816435,9275846301,9281637540,9284137056,9286750134,9287045316,9304786512,9305827461,9324815067,9328140756,9342675018,9356147802,9360154728,9368102754,9410736582,9418723056,9473680152,9476823015,9478165302,9483257061,9485702631,9504132678,9527406831,9538061724,9543281067,9543670812,9576210384,9601357482,9608371542,9618247053,9637458021,9643018752,9651037824,9652813074,9657841302,9672803514,9675104832,9681573204,9683104752,9704618532,9705843126,9706523481,9716243085,9740586231,9741253608,9748015632,9756230184,9768254310,9784561023,9785624301,9804631275,9804652317,9807213465,9824705613,9825640137,9834652071,9853024671,9865017243,9865403721,9876104253,9876124053}}

northwolves

n=8: {8, 3, {43165782, 43271856, 78324561}}
n=9: {9,28,{345918672,351987624,359841267,394675182,429715863,439516278,487256193,527394816,527498163,528714396,572493816,592681437,729564183,746318529,749258163,754932681,759142683,759823641,762491835,783942561,784196235,845691372,891357624,914863275,915786423,923165487,928163754,976825431}}

王守恩

你这个厉害了！！比 A003226——Automorphic numbers: m^2 ends with m. ——还厉害！！！

43165782^2=1863284735671524。
43271856^2=1872453521684736。
78324561^2=6134736855842721。
345918672^2=119659727638243584。
351987624^2=123895287449165376。
3175462089^2=10083559478676243921。
9876124053^2=97537826310245146809。

northwolves

$n^k$ 含$0-9$各$k$次(n也是0-9的全排列）：

A370667
Largest pandigital number whose n-th power contains each digit (0-9) exactly n times.

9876543210, 9876124053, 9863527104, 9846032571, 9847103256, 9247560381

COMMENTS
If an n-th power of a pandigital number k contains each digit (0-9) exactly n times, it implies that 10^(10 - 1/n) <= 9876543210, so n <= 185. It's easy to verify that no solutions exist for n=7 to 185.
LINKS
Table of n, a(n) for n=1..6.
EXAMPLE
a(4) = 9846032571 because it is the largest 10-digit number that contains each digit (0-9) exactly once and its 4th power 9398208429603554221689707364750715341681 contains each digit (0-9) exactly 4 times.

$k=3$: {74,{4680215379,4752360918,4765380219,4915280637,5063248197,5164738920,5382417906,5426370189,5429013678,5628130974,5679321048,5697841320,5762831940,5783610492,5786430129,5903467821,6019285734,6053147982,6095721483,6143720958,6158723094,6270841539,6295873401,6320184579,6351072984,6403859217,6429053817,6482357019,6487213905,6523710849,6705293418,6804793512,6829053147,6835274091,6857023419,7103895462,7105368294,7138509642,7251639804,7269184530,7295106438,7304291685,7403591286,7409658231,7459102386,7513968420,7538194206,7596214803,7619523048,7648130925,7891406235,7912308456,7945163082,7964025813,7980154263,8092146573,8269417305,8349570621,8469257031,8579462103,8769350412,9047813256,9075346128,9084371562,9216385407,9267083541,9306125487,9470156823,9483672015,9502148376,9563714028,9568042317,9754012836,9863527104}}

$k=4$: {13,{5702631489,7264103985,7602314895,7824061395,8105793624,8174035962,8304269175,8904623175,8923670541,9451360827,9785261403,9804753612,9846032571}}

$k=5$:  {8,{7351062489,8105632794,8401253976,8731945026,9164072385,9238750614,9615278340,9847103256}}

$k=6$: {6,{7025869314,7143258096,7931584062,8094273561,8920416357,9247560381}}