1572*8634=13572648

northwolves

{9,35271,{{5,20000051,100000255},{5,20000105,100000525},{5,20000501,100002505},{5,20000510,100002550},{5,20000519,100002595},{5,20000591,100002955},{5,20001005,100005025},{5,20001050,100005250},{5,20001059,100005295},{5,20001905,100009525},{5,20005001,100025005},{5,20005010,100025050},{5,20005019,100025095},{5,20005100,100025500},{5,20005190,100025950},{5,20005199,100025995},{5,20005901,100029505},{5,20005910,100029550},{5,20005919,100029595},{5,20005991,100029955},{5,20006513,100032565},{5,20006531,100032655},{5,20010005,100050025},{5,20010050,100050250},{5,20010059,100050295},{5,20010500,100052500},{5,20010590,100052950},{5,20010599,100052995},{5,20010653,100053265},{5,20013065,100065325},{5,20019005,100095025},{5,20019050,100095250},{5,20019059,100095295},{5,20019905,100099525},{5,20050001,100250005},{5,20050010,100250050},{5,20050019,100250095},{5,20050100,100250500},{5,20050190,100250950},{5,20050199,100250995},{5,20050613,100253065},{5,20051000,100255000},{5,20051306,100256530},{5,20051900,100259500},{5,20051990,100259950},{5,20051999,100259995},{5,20053061,100265305},{5,20053106,100265530},{5,20059001,100295005},{5,20059010,100295050},{5,20059019,100295095},{5,20059100,100295500},{5,20059190,100295950},{5,20059199,100295995},{5,20059901,100299505},{5,20059910,100299550},{5,20059919,100299595},{5,20059991,100299955},{5,20060513,100302565},{5,20060531,100302655},{5,20061053,100305265},{5,20061305,100306525},{5,20065013,100325065},{5,20065130,100325650},{5,20065139,100325695},{5,20065193,100325965},{5,20065301,100326505},{5,20065310,100326550},{5,20065319,100326595},{5,20065391,100326955},{5,20065913,100329565},{5,20065931,100329655},{5,20100005,100500025},{5,20100050,100500250},{5,20100059,100500295},{5,20100500,100502500},{5,20100590,100502950},{5,20100599,100502995},{5,20100653,100503265},{5,20105000,100525000},{5,20105306,100526530},{5,20105900,100529500},{5,20105990,100529950},{5,20105999,100529995},{5,20106053,100530265},{5,20106530,100532650},{5,20106539,100532695},{5,20106593,100532965},{5,20130065,100650325},{5,20130506,100652530},{5,20130605,100653025},{5,20130650,100653250},{5,20130659,100653295},{5,20139065,100695325},{8,12603047,100824376},{8,12604703,100837624},{5,20190005,100950025},{5,20190050,100950250},{5,20190059,100950295},{5,20190500,100952500},{...................},{9,98903511,890131599},{9,98910351,890193159},{9,98911035,890199315},{9,98988021,890892189},{9,98988102,890892918},{9,99216855,892951695},{9,99218466,892966194},{9,99220788,892987092},{9,99220878,892987902},{9,99221868,892996812},{9,99243801,893194209},{9,99243810,893194290},{9,99332748,893994732},{9,99436851,894931659},{9,99483255,895349295},{9,99548325,895934925},{9,99552168,895969512},{9,99643887,896794983},{9,99768501,897916509},{9,99768510,897916590},{9,99777483,897997347},{9,99777528,897997752},{9,99868851,898819659},{9,99881325,898931925},{9,99882135,898939215},{9,99883755,898953795},{9,99884601,898961409},{9,99884610,898961490},{9,99886851,898981659},{9,99887931,898991379},{9,99888012,898992108},{9,99888750,898998750},{9,99924381,899319429},{9,99976851,899791659},{9,99988461,899896149},{9,99988875,899899875}}}

northwolves

若 $A*B=C$,不限制A 为2-9：

1. For[n = 1, n <= 7, n++, s = {};
   
2. For[y = 10^(n - 1), y < 10^n, y++, d = Divisors[y];
   
3.   For[r = 2, r <= (Length@d + 1)/2, r++, k = d[[r]];
   
4.    If[k^2 <= y, x = y/k;
   
5. a = SortBy[Tally[Join[IntegerDigits[x], IntegerDigits[k]]], First];
   
6. b = SortBy[Tally[IntegerDigits[y]], First];
   
7.     If[a == b, AppendTo[s, {k, x, y}]]]]]; Print[{n, Length@s, s}]]

复制代码

northwolves

{1,0,{}}
{2,0,{}}
{3,3,{{6,21,126},{3,51,153},{8,86,688}}}
{4,15,{{6,201,1206},{5,251,1255},{6,210,1260},{21,60,1260},{15,93,1395},{35,41,1435},{3,501,1503},{3,510,1530},{30,51,1530},{21,87,1827},{27,81,2187},{9,351,3159},{8,473,3784},{8,860,6880},{80,86,6880}}}
{5,98,{{51,201,10251},{5,2051,10255},{26,401,10426},{21,501,10521},{5,2105,10525},{15,705,10575},{9,1251,11259},{84,141,11844},{8,1481,11848},{6,2001,12006},{6,2010,12060},{60,201,12060},{3,4128,12384},{5,2501,12505},{51,246,12546},{5,2510,12550},{50,251,12550},{5,2519,12595},{6,2100,12600},{21,600,12600},{60,210,12600},{6,2127,12762},{3,4281,12843},{5,2591,12955},{14,926,12964},{41,323,13243},{3,4515,13545},{15,930,13950},{93,150,13950},{35,401,14035},{35,410,14350},{41,350,14350},{3,5001,15003},{3,5010,15030},{30,501,15030},{6,2541,15246},{3,5100,15300},{30,510,15300},{51,300,15300},{3,5145,15435},{24,651,15624},{9,1755,15795},{6,2712,16272},{75,231,17325},{2,8714,17428},{47,371,17437},{2,8741,17482},{81,225,18225},{65,281,18265},{21,870,18270},{87,210,18270},{21,906,19026},{21,915,19215},{3,7125,21375},{86,251,21586},{3,7251,21753},{27,810,21870},{81,270,21870},{5,5021,25105},{35,725,25375},{47,542,25474},{5,5102,25510},{42,678,28476},{32,926,29632},{9,3501,31509},{9,3510,31590},{90,351,31590},{53,635,33655},{36,936,33696},{63,585,36855},{8,4730,37840},{80,473,37840},{87,435,37845},{8,4973,39784},{65,641,41665},{89,482,42898},{6,7446,44676},{54,846,45684},{65,704,45760},{84,546,45864},{57,834,47538},{78,624,48672},{59,845,49855},{9,5751,51759},{8,6521,52168},{63,855,53865},{9,6255,56295},{65,875,56875},{68,926,62968},{65,983,63895},{9,7461,67149},{72,936,67392},{75,906,67950},{8,8600,68800},{80,860,68800},{86,800,68800},{9,7911,71199},{9,8775,78975}}}
{6,596,{{5,20051,100255},{5,20105,100525},{11,9209,101299},{51,2001,102051},{5,20501,102505},{51,2010,102510},{201,510,102510},{41,2501,102541},{5,20510,102550},{50,2051,102550},{5,20519,102595},{21,4902,102942},{5,20591,102955},{96,1083,103968},{26,4001,104026},{26,4010,104260},{260,401,104260},{8,13046,104368},{21,5001,105021},{5,21005,105025},{15,7005,105075},{21,5010,105210},{210,501,105210},{5,21050,105250},{50,2105,105250},{51,2064,105264},{204,516,105264},{5,21059,105295},{51,2073,105723},{15,7050,105750},{150,705,105750},{63,1701,107163},{29,3701,107329},{6,18021,108126},{135,801,108135},{18,6012,108216},{6,18102,108612},{18,6048,108864},{5,21905,109525},{158,701,110758},{62,1814,112468},{9,12501,112509},{9,12510,112590},{90,1251,112590},{14,8162,114268},{152,761,115672},{59,1961,115699},{18,6471,116478},{6,19416,116496},{161,725,116725},{96,1218,116928},{167,701,117067},{8,14801,118408},{84,1410,118440},{141,840,118440},{8,14810,118480},{80,1481,118480},{75,1581,118575},{6,19821,118926},{8,14981,119848},{6,20001,120006},{6,20010,120060},{60,2001,120060},{3,40128,120384},{6,20100,120600},{60,2010,120600},{201,600,120600},{6,20127,120762},{3,40281,120843},{6,20181,121086},{56,2171,121576},{15,8121,121815},{26,4721,122746},{47,2612,122764},{3,41028,123084},{231,534,123354},{38,3251,123538},{3,41280,123840},{30,4128,123840},{3,41298,123894},{281,443,124483},{84,1482,124488},{51,2442,124542},{14,8927,124978},{5,25001,125005},{5,25010,125050},{50,2501,125050},{5,25019,125095},{152,824,125248},{231,543,125433},{51,2460,125460},{204,615,125460},{246,510,125460},{5,25100,125500},{50,2510,125500},{251,500,125500},{5,25190,125950},{50,2519,125950},{5,25199,125995},{6,21000,126000},{21,6000,126000},{60,2100,126000},{210,600,126000},{201,627,126027},{6,21018,126108},{261,486,126846},{47,2711,127417},{6,21270,127620},{60,2127,127620},{3,42801,128403},{3,42810,128430},{30,4281,128430},{3,42981,128943},{9,14382,129438},{5,25901,129505},{14,9251,129514},{5,25910,129550},{50,2591,129550},{6,21594,129564},{5,25919,129595},{14,9260,129640},{140,926,129640},{179,725,129775},{5,25991,129955},{311,422,131242},{41,3230,132430},{323,410,132430},{5,26513,132565},{21,6315,132615},{5,26531,132655},{315,423,133245},{41,3275,134275},{317,425,134725},{3,45015,135045},{3,45150,135450},{30,4515,135450},{231,588,135828},{351,387,135837},{215,635,136525},{3,45618,136854},{146,938,136948},{8,17348,138784},{15,9300,139500},{93,1500,139500},{150,930,139500},{48,2913,139824},{35,4001,140035},{35,4010,140350},{350,401,140350},{45,3141,141345},{2,71489,142978},{35,4100,143500},{41,3500,143500},{350,410,143500},{3,47913,143739},{3,47931,143793},{53,2741,145273},{351,414,145314},{41,3545,145345},{3,48561,145683},{317,461,146137},{24,6105,146520},{156,942,146952},{36,4149,149364},{2,74891,149782},{3,50001,150003},{3,50010,150030},{30,5001,150030},{6,25041,150246},{3,50100,150300},{30,5010,150300},{300,501,150300},{3,50145,150435},{6,25104,150624},{26,5801,150826},{21,7251,152271},{6,25401,152406},{6,25410,152460},{60,2541,152460},{251,608,152608},{261,585,152685},{6,25491,152946},{3,51000,153000},{30,5100,153000},{51,3000,153000},{300,510,153000},{356,431,153436},{3,51450,154350},{30,5145,154350},{27,5751,155277},{24,6501,156024},{24,6510,156240},{240,651,156240},{269,581,156289},{5,31265,156325},{165,951,156915},{9,17550,157950},{90,1755,157950},{81,1953,158193},{6,27012,162072},{26,6251,162526},{6,27120,162720},{60,2712,162720},{176,926,162976},{5,32651,163255},{17,9635,163795},{3,54618,163854},{396,414,163944},{3,54861,164583},{8,20651,165208},{8,21065,168520},{98,1751,171598},{71,2426,172246},{6,28731,172386},{221,782,172822},{75,2310,173250},{231,750,173250},{75,2319,173925},{2,87014,174028},{2,87041,174082},{2,87104,174208},{2,87140,174280},{20,8714,174280},{2,87149,174298},{47,3710,174370},{371,470,174370},{47,3719,174793},{2,87401,174802},{2,87410,174820},{20,8741,174820},{2,87491,174982},{231,759,175329},{65,2711,176215},{2,89147,178294},{2,89471,178942},{75,2391,179325},{2,89714,179428},{2,89741,179482},{225,801,180225},{201,897,180297},{21,8601,180621},{65,2801,182065},{81,2250,182250},{225,810,182250},{65,2810,182650},{281,650,182650},{21,8700,182700},{87,2100,182700},{210,870,182700},{2,91487,182974},{86,2141,184126},{216,864,186624},{9,20781,187029},{9,21078,189702},{2,94871,189742},{21,9060,190260},{210,906,190260},{21,9087,190827},{21,9105,191205},{21,9150,192150},{210,915,192150},{9,21375,192375},{89,2165,192685},{21,9177,192717},{327,591,193257},{395,491,193945},{21,9249,194229},{2,98714,197428},{2,98741,197482},{275,719,197725},{252,801,201852},{255,807,205785},{23,9017,207391},{26,8024,208624},{3,70125,210375},{81,2601,210681},{3,70251,210753},{216,981,211896},{81,2628,212868},{3,71025,213075},{341,626,213466},{3,71250,213750},{30,7125,213750},{9,23751,213759},{51,4206,214506},{86,2501,215086},{51,4224,215424},{41,5255,215455},{86,2510,215860},{251,860,215860},{323,671,216733},{3,72501,217503},{3,72510,217530},{30,7251,217530},{321,678,217638},{42,5187,217854},{248,881,218488},{27,8100,218700},{81,2700,218700},{270,810,218700},{42,5322,223524},{269,842,226498},{276,822,226872},{26,8726,226876},{27,8424,227448},{248,926,229648},{9,25731,231579},{71,3263,231673},{338,692,233896},{42,5637,236754},{86,2753,236758},{9,26325,236925},{6,39828,238968},{6,40251,241506},{461,524,241564},{71,3425,243175},{422,581,245182},{42,5844,245448},{6,41025,246150},{92,2684,246928},{5,50021,250105},{5,50102,250510},{5,50201,251005},{5,50210,251050},{50,5021,251050},{5,50219,251095},{296,851,251896},{35,7250,253750},{350,725,253750},{47,5420,254740},{470,542,254740},{5,51002,255010},{5,51020,255100},{50,5102,255100},{5,51263,256315},{42,6105,256410},{41,6254,256414},{9,28755,258795},{27,9585,258795},{5,51902,259510},{323,806,260338},{3,87126,261378},{3,87261,261783},{284,926,262984},{437,602,263074},{5,52631,263155},{36,7326,263736},{72,3663,263736},{62,4307,267034},{39,6882,268398},{93,2886,268398},{32,8729,279328},{78,3612,281736},{89,3182,283198},{32,8864,283648},{489,582,284598},{42,6780,284760},{420,678,284760},{8,35672,285376},{468,612,286416},{6,47829,286974},{38,7562,287356},{6,48279,289674},{42,6897,289674},{3,97125,291375},{3,97251,291753},{9,32625,293625},{5,59021,295105},{5,59102,295510},{32,9260,296320},{320,926,296320},{47,6329,297463},{8,37229,297832},{431,707,304717},{83,3701,307183},{431,725,312475},{5,62513,312565},{5,62531,312655},{321,975,312975},{9,34911,314199},{71,4433,314743},{9,35001,315009},{9,35010,315090},{90,3501,315090},{35,9014,315490},{534,591,315594},{5,63125,315625},{9,35100,315900},{90,3510,315900},{351,900,315900},{5,63251,316255},{351,909,319059},{336,951,319536},{5,65123,325615},{5,65231,326155},{524,623,326452},{39,8421,328419},{86,3824,328864},{342,963,329346},{356,926,329656},{93,3615,336195},{53,6350,336550},{530,635,336550},{36,9360,336960},{360,936,336960},{392,863,338296},{41,8324,341284},{533,641,341653},{8,42836,342688},{8,43286,346288},{75,4623,346725},{366,948,346968},{93,3741,347913},{62,5693,352966},{9,39555,355995},{369,981,361989},{392,926,362992},{533,686,365638},{8,46013,368104},{63,5850,368550},{585,630,368550},{48,7683,368784},{381,969,369189},{383,971,371893},{8,46733,373864},{51,7356,375156},{51,7365,375615},{96,3927,376992},{8,47300,378400},{80,4730,378400},{473,800,378400},{431,878,378418},{87,4350,378450},{435,870,378450},{9,42381,381429},{432,891,384912},{89,4325,384925},{465,831,386415},{83,4709,390847},{593,662,392566},{42,9363,393246},{9,43713,393417},{9,43875,394875},{8,49730,397840},{80,4973,397840},{8,49973,399784},{44,9203,404932},{446,908,404968},{491,845,414895},{75,5541,415575},{65,6401,416065},{9,46251,416259},{65,6410,416650},{641,650,416650},{468,891,416988},{47,8921,419287},{89,4820,428980},{482,890,428980},{464,926,429664},{8,54473,435784},{53,8294,439582},{47,9425,442975},{6,74460,446760},{60,7446,446760},{6,74496,446976},{476,941,447916},{6,74946,449676},{99,4545,449955},{56,8048,450688},{8,56471,451768},{54,8460,456840},{540,846,456840},{8,57146,457168},{65,7040,457600},{650,704,457600},{84,5460,458640},{546,840,458640},{72,6426,462672},{86,5408,465088},{96,4854,465984},{83,5645,468535},{57,8340,475380},{570,834,475380},{57,8349,475893},{6,79482,476892},{78,6240,486720},{624,780,486720},{9,54288,488592},{549,891,489159},{545,899,489955},{69,7104,490176},{6,81948,491688},{9,54873,493857},{92,5384,495328},{6,82947,497682},{59,8450,498550},{590,845,498550},{57,9051,515907},{681,759,516879},{9,57501,517509},{9,57510,517590},{90,5751,517590},{95,5471,519745},{8,65021,520168},{8,65102,520816},{8,65201,521608},{8,65210,521680},{80,6521,521680},{92,5726,526792},{572,926,529672},{93,5703,530379},{57,9321,531297},{96,5583,535968},{563,953,536539},{63,8550,538650},{630,855,538650},{57,9645,549765},{588,951,559188},{9,62550,562950},{90,6255,562950},{6,94152,564912},{657,864,567648},{65,8750,568750},{650,875,568750},{8,71456,571648},{78,7356,573768},{68,8657,588676},{63,9711,611793},{71,8618,611878},{98,6251,612598},{65,9461,614965},{8,77216,617728},{9,68751,618759},{86,7253,623758},{68,9260,629680},{680,926,629680},{83,7625,632875},{65,9830,638950},{650,983,638950},{66,9843,649638},{8,82661,661288},{69,9651,665919},{86,7766,667876},{9,74601,671409},{9,74610,671490},{90,7461,671490},{71,9464,671944},{72,9360,673920},{720,936,673920},{89,7628,678892},{75,9060,679500},{750,906,679500},{71,9689,687919},{8,86000,688000},{80,8600,688000},{86,8000,688000},{800,860,688000},{72,9621,692712},{72,9684,697248},{9,78021,702189},{9,78102,702918},{96,7401,710496},{9,79011,711099},{9,79101,711909},{9,79110,711990},{90,7911,711990},{9,79551,715959},{9,79911,719199},{788,926,729688},{765,963,736695},{843,876,738468},{8,92741,741928},{776,992,769792},{8,96737,773896},{8,97367,778936},{8,97862,782896},{9,87255,785295},{875,902,789250},{825,957,789525},{9,87750,789750},{90,8775,789750},{9,87921,791289},{855,927,792585},{807,984,794088},{86,9287,798682},{891,909,809919},{93,8709,809937},{894,906,809964},{858,951,815958},{896,926,829696},{891,945,841995},{96,8958,859968},{99,9081,899019},{95,9863,936985},{953,986,939658},{98,9806,960988}}}

northwolves

{7,3626,{{5,200051,1000255},{5,200105,1000525},{501,2001,1002501},{5,200501,1002505},{5,200510,1002550},{50,20051,1002550},{5,200519,1002595},{5,200591,1002955},{251,4001,1004251},{5,201005,1005025},{201,5001,1005201},{5,201050,1005250},{50,20105,1005250},{5,201059,1005295},{126,8001,1008126},{5,201905,1009525},{11,92009,1012099},{11,92018,1012198},{11,92027,1012297},{11,92036,1012396},{11,92045,1012495},{81,12501,1012581},{11,92054,1012594},{11,92063,1012693},{11,92072,1012792},{11,92081,1012891},{11,92090,1012990},{110,9209,1012990},{975,1041,1014975},{168,6051,1016568},{635,1601,1016635},{782,1301,1017382},{11,92702,1019722},{51,20001,1020051},{51,20010,1020510},{510,2001,1020510},{501,2037,1020537},{906,1128,1021968},{11,92909,1021999},{401,2555,1024555},{251,4082,1024582},{21,48804,1024884},{............................},{99,80901,8009199},{98,81806,8016988},{89,90446,8049694},{93,86703,8063379},{89,90671,8069719},{894,9051,8091594},{891,9090,8099190},{909,8910,8099190},{93,87090,8099370},{930,8709,8099370},{894,9060,8099640},{906,8940,8099640},{95,85301,8103595},{89,91382,8132998},{96,84801,8140896},{95,85841,8154895},{851,9584,8155984},{858,9510,8159580},{951,8580,8159580},{987,8301,8193087},{944,8681,8194864},{99,82818,8198982},{995,8261,8219695},{921,8925,8219925},{839,9821,8239819},{98,84326,8263948},{89,93182,8293198},{96,86421,8296416},{896,9260,8296960},{926,8960,8296960},{858,9672,8298576},{93,89241,8299413},{9,924381,8319429},{86,97316,8369176},{873,9612,8391276},{863,9725,8392675},{89,94325,8394925},{99,85041,8419059},{891,9450,8419950},{945,8910,8419950},{884,9569,8458996},{965,8804,8495860},{905,9458,8559490},{96,89580,8599680},{960,8958,8599680},{996,8751,8715996},{95,91781,8719195},{888,9876,8769888},{9,976851,8791659},{9,988461,8896149},{9,988875,8899875},{909,9801,8909109},{92,97688,8987296},{99,90810,8990190},{990,9081,8990190},{909,9891,8990919},{96,93684,8993664},{96,94026,9026496},{99,91305,9039195},{951,9642,9169542},{941,9755,9179455},{95,98630,9369850},{950,9863,9369850},{953,9860,9396580},{986,9530,9396580},{978,9642,9429876},{984,9627,9472968},{96,99552,9556992},{98,98060,9609880},{980,9806,9609880},{98,98366,9639868},{992,9776,9697792}}}

王守恩

northwolves 发表于 2025-10-30 14:30
{1,0,{}}
{2,0,{}}
{3,3,{{6,21,126},{3,51,153},{8,86,688}}}

A×B=C。A=3。左边的数码=右边的数码。

3×51=153。
2位数有1个。

3×501=1503。
3×510=1530。
3位数有2个。

3×4128=12384。
3×4281=12843。
3×4515=13545。——A=3。左边的数码=右边的数码。
3×5001=15003。
3×5010=15030。
3×5100=15300。
3×5145=15435。
3×7125=21375。
3×7251=21753。
4位数有9个。

3×40128=120384。
3×40281=120843。
3×41028=123084。
3×41280=123840。
3×41298=123894。——A=3。左边的数码=右边的数码。
3×42801=128403。
3×42810=128430。
3×42981=128943。
3×45015=135045。
3×45150=135450。——A=3。左边的数码=右边的数码。
3×45618=136854。
3×47913=143739。
3×47931=143793。
3×48561=145683。
......
1位数有0个。2位数有1个。3位数有2个。4位数有9个。5位数有??个。6位数有???个。7位数有????个。——0, 1, 2, 9, 32, ...... ——OEIS没有这串数。——我来不了。

northwolves

{8,22704,{{5,2000051,10000255},{5,2000105,10000525},{5,2000501,10002505},{5,2000510,10002550},{50,200051,10002550},{5,2000519,10002595},{5,2000591,10002955},{5,2001005,10005025},{5,2001050,10005250},{50,200105,10005250},{5,2001059,10005295},{5,2001905,10009525},{11,910109,10011199},{323,31001,10013323},{626,16001,10016626},{99,101205,10019295},{11,910901,10019911},{501,20001,10020501},{5,2005001,10025005},{501,20010,10025010},{2001,5010,10025010},{5,2005010,10025050},{50,200501,10025050},{5,2005019,10025095},{401,25001,10025401},{5,2005100,10025500},{50,200510,10025500},{500,20051,10025500},{5,2005190,10025950},{50,200519,10025950},{5,2005199,10025995},{5,2005901,10029505},{5,2005910,10029550},{50,200591,10029550},{5,2005919,10029595},{5,2005991,10029955},{5,2006513,10032565},{5,2006531,10032655},{77,130301,10033177},{251,40001,10040251},{96,104601,10041696},{251,40010,10042510},{2510,4001,10042510},{41,245006,10045246},{5,2010005,10050025},{201,50001,10050201},{5,2010050,10050250},{50,201005,10050250},{5,2010059,10050295},{201,50010,10052010},{2010,5001,10052010},{501,20064,10052064},{2004,5016,10052064},{5,2010500,10052500},{50,201050,10052500},{500,20105,10052500},{5,2010590,10052950},{50,201059,10052950},{5,2010599,10052995},{5,2010653,10053265},{401,25073,10054273},{387,26001,10062387},{5,2013065,10065325},{95,106073,10076935},{126,80001,10080126},{12,840102,10081224},{126,80010,10081260},{1260,8001,10081260},{21,480102,10082142},{12,840201,10082412},{21,480201,10084221},{348,29001,10092348},{206,49001,10094206},{48,210309,10094832},{5,2019005,10095025},{5,2019050,10095250},{50,201905,10095250},{5,2019059,10095295},{51,198003,10098153},{5,2019905,10099525},{..................},{9489,9501,90154989},{96,940260,90264960},{960,94026,90264960},{95,950273,90275935},{921,98034,90289314},{981,92043,90294183},{99,913050,90391950},{990,91305,90391950},{93,974058,90587394},{9401,9647,90691447},{969,93801,90893169},{939,96801,90896139},{951,95601,90916551},{9179,9905,90917995},{9374,9701,90937174},{9407,9668,90946876},{95,958301,91038595},{96,948801,91084896},{9501,9588,91095588},{968,94211,91196248},{986,92501,91205986},{921,99033,91209393},{975,93741,91397475},{971,94145,91414795},{95,965213,91695235},{951,96420,91695420},{9510,9642,91695420},{9561,9591,91699551},{941,97550,91794550},{9410,9755,91794550},{99,928071,91879029},{9501,9678,91950678},{972,94617,91967724},{98,942305,92345890},{98,942341,92349418},{99,933327,92399373},{941,98234,92438194},{99,934281,92493819},{9521,9725,92591725},{963,96255,92693565},{98,946427,92749846},{9426,9852,92864952},{9461,9821,92916481},{96,968421,92968416},{9428,9863,92988364},{956,97301,93019756},{983,94709,93098947},{95,982043,93294085},{983,95024,93408592},{993,94155,93495915},{9513,9837,93579381},{95,986300,93698500},{950,98630,93698500},{9500,9863,93698500},{95,987353,93798535},{981,95634,93816954},{95,989063,93960985},{953,98600,93965800},{986,95300,93965800},{9530,9860,93965800},{947,99236,93976492},{96,981426,94216896},{978,96420,94298760},{9642,9780,94298760},{9534,9897,94357998},{95,993473,94379935},{945,99873,94379985},{96,984342,94496832},{9546,9912,94619952},{984,96270,94729680},{9627,9840,94729680},{9614,9857,94765198},{9617,9854,94765918},{96,987483,94798368},{98,968234,94886932},{987,96141,94891167},{99,958545,94895955},{9705,9834,95438970},{96,995520,95569920},{960,99552,95569920},{98,975254,95574892},{969,98751,95689719},{98,978305,95873890},{9729,9855,95879295},{96,998958,95899968},{96,999552,95956992},{9692,9905,95999260},{98,980600,96098800},{980,98060,96098800},{9800,9806,96098800},{965,99803,96309895},{98,983660,96398680},{980,98366,96398680},{9702,9936,96399072},{98,984641,96494818},{98,985631,96591838},{992,97760,96977920},{9776,9920,96977920},{974,99731,97137994},{98,991751,97191598},{986,99287,97896982},{99,998181,98819919}}}

northwolves

王守恩 发表于 2025-10-30 16:48
A×B=C。A=3。左边的数码=右边的数码。

3×51=153。

1. For[n=1,n<=6,n++,s={};For[y=10^n+2,y<3*10^n,y+=3,x=y/3;a=SortBy[Tally[Join[IntegerDigits[x],{3}]],First];b=SortBy[Tally[IntegerDigits[y]],First];If[a==b,AppendTo[s,{3,x,y}]]];Print[{n,Length@s,s}]]

复制代码

{1,0,{}}
{2,1,{{3,51,153}}}
{3,2,{{3,501,1503},{3,510,1530}}}
{4,9,{{3,4128,12384},{3,4281,12843},{3,4515,13545},{3,5001,15003},{3,5010,15030},{3,5100,15300},{3,5145,15435},{3,7125,21375},{3,7251,21753}}}
{5,32,{{3,40128,120384},{3,40281,120843},{3,41028,123084},{3,41280,123840},{3,41298,123894},{3,42801,128403},{3,42810,128430},{3,42981,128943},{3,45015,135045},{3,45150,135450},{3,45618,136854},{3,47913,143739},{3,47931,143793},{3,48561,145683},{3,50001,150003},{3,50010,150030},{3,50100,150300},{3,50145,150435},{3,51000,153000},{3,51450,154350},{3,54618,163854},{3,54861,164583},{3,70125,210375},{3,70251,210753},{3,71025,213075},{3,71250,213750},{3,72501,217503},{3,72510,217530},{3,87126,261378},{3,87261,261783},{3,97125,291375},{3,97251,291753}}}
{6,138,{{3,350511,1051533},{3,351051,1053153},{3,351105,1053315},{3,379113,1137339},{3,379131,1137393},{3,379311,1137933},{3,385611,1156833},{3,400128,1200384},{3,400281,1200843},{3,401028,1203084},{3,401280,1203840},{3,401298,1203894},{3,402801,1208403},{3,402810,1208430},{3,402981,1208943},{3,410028,1230084},{3,410280,1230840},{3,410298,1230894},{3,412800,1238400},{3,412980,1238940},{3,412998,1238994},{3,414528,1243584},{3,428001,1284003},{3,428010,1284030},{3,428100,1284300},{3,428145,1284435},{3,429801,1289403},{3,429810,1289430},{3,429981,1289943},{3,441528,1324584},{3,442815,1328445},{3,450015,1350045},{3,450150,1350450},{3,451428,1354284},{3,451500,1354500},{3,452814,1358442},{3,456018,1368054},{3,456180,1368540},{3,456198,1368594},{3,475125,1425375},{3,475251,1425753},{3,479013,1437039},{3,479031,1437093},{3,479103,1437309},{3,479130,1437390},{3,479301,1437903},{3,479310,1437930},{3,479913,1439739},{3,479931,1439793},{3,485601,1456803},{3,485610,1456830},{3,489651,1468953},{3,497913,1493739},{3,497931,1493793},{3,498561,1495683},{3,500001,1500003},{3,500010,1500030},{3,500100,1500300},{3,500145,1500435},{3,501000,1503000},{3,501450,1504350},{3,503511,1510533},{3,510000,1530000},{3,510351,1531053},{3,511035,1533105},{3,512475,1537425},{3,514428,1543284},{3,514500,1543500},{3,524751,1574253},{3,528144,1584432},{3,538611,1615833},{3,546018,1638054},{3,546180,1638540},{3,546198,1638594},{3,548601,1645803},{3,548610,1645830},{3,549861,1649583},{3,565179,1695537},{3,648951,1946853},{3,655179,1965537},{3,700125,2100375},{3,700251,2100753},{3,701025,2103075},{3,701250,2103750},{3,702501,2107503},{3,702510,2107530},{3,710025,2130075},{3,710250,2130750},{3,712428,2137284},{3,712500,2137500},{3,712824,2138472},{3,714525,2143575},{3,715245,2145735},{3,724128,2172384},{3,724281,2172843},{3,724515,2173545},{3,725001,2175003},{3,725010,2175030},{3,725100,2175300},{3,725145,2175435},{3,728124,2184372},{3,728241,2184723},{3,779253,2337759},{3,779325,2337975},{3,785256,2355768},{3,785526,2356578},{3,785625,2356875},{3,786255,2358765},{3,812427,2437281},{3,812724,2438172},{3,824127,2472381},{3,824271,2472813},{3,827124,2481372},{3,827241,2481723},{3,828456,2485368},{3,828546,2485638},{3,845628,2536884},{3,854628,2563884},{3,870126,2610378},{3,870261,2610783},{3,871026,2613078},{3,871260,2613780},{3,872601,2617803},{3,872610,2617830},{3,897126,2691378},{3,897261,2691783},{3,946482,2839446},{3,964482,2893446},{3,970125,2910375},{3,970251,2910753},{3,971025,2913075},{3,971250,2913750},{3,972501,2917503},{3,972510,2917530},{3,987126,2961378},{3,987261,2961783},{3,997125,2991375},{3,997251,2991753}}}

northwolves

{7,817,{{3,3460812,10382436},{3,3460821,10382463},{3,3461082,10383246},{3,3461208,10383624},{3,3462081,10386243},{3,3462108,10386324},{3,3500511,10501533},{3,3501051,10503153},{3,3501105,10503315},{3,3505011,10515033},{3,3505101,10515303},{3,3505110,10515330},{3,3510051,10530153},{3,3510105,10530315},{3,3510501,10531503},{3,3510510,10531530},{3,3511005,10533015},{3,3511050,10533150},{3,3751125,11253375},{3,3751251,11253753},{3,3752511,11257533},{3,3790113,11370339},{3,3790131,11370393},{3,3790311,11370933},{3,3791013,11373039},{3,3791031,11373093},{3,3791103,11373309},{3,3791130,11373390},{3,3791301,11373903},{3,3791310,11373930},{3,3793011,11379033},{3,3793101,11379303},{3,3793110,11379330},{3,3799113,11397339},{3,3799131,11397393},{3,3799311,11397933},{3,3856011,11568033},{3,3856101,11568303},{3,3856110,11568330},{3,3896511,11689533},{3,3979113,11937339},{3,3979131,11937393},{3,3979311,11937933},{3,3985611,11956833},{................................},{3,9700125,29100375},{3,9700251,29100753},{3,9701025,29103075},{3,9701250,29103750},{3,9702501,29107503},{3,9702510,29107530},{3,9710025,29130075},{3,9710250,29130750},{3,9712428,29137284},{3,9712500,29137500},{3,9712824,29138472},{3,9714525,29143575},{3,9715245,29145735},{3,9724128,29172384},{3,9724281,29172843},{3,9724515,29173545},{3,9725001,29175003},{3,9725010,29175030},{3,9725100,29175300},{3,9725145,29175435},{3,9728124,29184372},{3,9728241,29184723},{3,9779253,29337759},{3,9779325,29337975},{3,9785256,29355768},{3,9785526,29356578},{3,9785625,29356875},{3,9786255,29358765},{3,9812427,29437281},{3,9812724,29438172},{3,9824127,29472381},{3,9824271,29472813},{3,9827124,29481372},{3,9827241,29481723},{3,9828456,29485368},{3,9828546,29485638},{3,9845628,29536884},{3,9854628,29563884},{3,9870126,29610378},{3,9870261,29610783},{3,9871026,29613078},{3,9871260,29613780},{3,9872601,29617803},{3,9872610,29617830},{3,9897126,29691378},{3,9897261,29691783},{3,9946482,29839446},{3,9964482,29893446},{3,9970125,29910375},{3,9970251,29910753},{3,9971025,29913075},{3,9971250,29913750},{3,9972501,29917503},{3,9972510,29917530},{3,9987126,29961378},{3,9987261,29961783},{3,9997125,29991375},{3,9997251,29991753}}}

{8,5005,{{3,34051128,102153384},{3,34051281,102153843},{3,34052811,102158433},{3,34105128,102315384},{3,34105281,102315843},{3,34110528,102331584},{3,34112805,102338415},{3,34128051,102384153},{3,34128105,102384315},{3,34280511,102841533},{3,34281051,102843153},{3,34281105,102843315},{3,34505115,103515345},{3,34505151,103515453},{3,34510515,103531545},{3,34511505,103534515},{3,34515051,103545153},{3,34515105,103545315},{3,34600812,103802436},{3,34600821,103802463},{3,34601082,103803246},{3,34601208,103803624},{3,34602081,103806243},{3,34602108,103806324},{3,34608012,103824036},{3,34608021,103824063},{3,34608102,103824306},{3,34608120,103824360},{3,34608201,103824603},{3,34608210,103824630},{3,34609812,103829436},{3,34609821,103829463},{3,34610082,103830246},{3,34610208,103830624},{3,34610802,103832406},{3,34610820,103832460},{3,34610982,103832946},{3,34612008,103836024},{3,34612080,103836240},{3,34612098,103836294},{3,34620081,103860243},{3,34620108,103860324},{3,34620801,103862403},{3,34620810,103862430},{3,34620981,103862943},{3,34621008,103863024},{3,34621080,103863240},{3,34621098,103863294},{..................................},{3,99700125,299100375},{3,99700251,299100753},{3,99701025,299103075},{3,99701250,299103750},{3,99702501,299107503},{3,99702510,299107530},{3,99710025,299130075},{3,99710250,299130750},{3,99712428,299137284},{3,99712500,299137500},{3,99712824,299138472},{3,99714525,299143575},{3,99715245,299145735},{3,99724128,299172384},{3,99724281,299172843},{3,99724515,299173545},{3,99725001,299175003},{3,99725010,299175030},{3,99725100,299175300},{3,99725145,299175435},{3,99728124,299184372},{3,99728241,299184723},{3,99779253,299337759},{3,99779325,299337975},{3,99785256,299355768},{3,99785526,299356578},{3,99785625,299356875},{3,99786255,299358765},{3,99812427,299437281},{3,99812724,299438172},{3,99824127,299472381},{3,99824271,299472813},{3,99827124,299481372},{3,99827241,299481723},{3,99828456,299485368},{3,99828546,299485638},{3,99845628,299536884},{3,99854628,299563884},{3,99870126,299610378},{3,99870261,299610783},{3,99871026,299613078},{3,99871260,299613780},{3,99872601,299617803},{3,99872610,299617830},{3,99897126,299691378},{3,99897261,299691783},{3,99946482,299839446},{3,99964482,299893446},{3,99970125,299910375},{3,99970251,299910753},{3,99971025,299913075},{3,99971250,299913750},{3,99972501,299917503},{3,99972510,299917530},{3,99987126,299961378},{3,99987261,299961783},{3,99997125,299991375},{3,99997251,299991753}}}

{9,31732,{{3,340051128,1020153384},{3,340051281,1020153843},{3,340052811,1020158433},{3,340105128,1020315384},{3,340105281,1020315843},{3,340110528,1020331584},{3,340112805,1020338415},{3,340128051,1020384153},{3,340128105,1020384315},{3,340280511,1020841533},{3,340281051,1020843153},{3,340281105,1020843315},{3,340501128,1021503384},{3,340501281,1021503843},{3,340502811,1021508433},{3,340510128,1021530384},{3,340510281,1021530843},{3,340511028,1021533084},{3,340511280,1021533840},{3,340511298,1021533894},{3,340512801,1021538403},{3,340512810,1021538430},{3,340512981,1021538943},{3,340528011,1021584033},{3,340528101,1021584303},{3,340528110,1021584330},{3,340529811,1021589433},{3,340611288,1021833864},{3,340612881,1021838643},{3,340628811,1021886433},{3,340781226,1022343678},{3,340781262,1022343786},{3,340782126,1022346378},{3,340782261,1022346783},{3,340782612,1022347836},{3,340782621,1022347863},{......................},{3,999779253,2999337759},{3,999779325,2999337975},{3,999785256,2999355768},{3,999785526,2999356578},{3,999785625,2999356875},{3,999786255,2999358765},{3,999812427,2999437281},{3,999812724,2999438172},{3,999824127,2999472381},{3,999824271,2999472813},{3,999827124,2999481372},{3,999827241,2999481723},{3,999828456,2999485368},{3,999828546,2999485638},{3,999845628,2999536884},{3,999854628,2999563884},{3,999870126,2999610378},{3,999870261,2999610783},{3,999871026,2999613078},{3,999871260,2999613780},{3,999872601,2999617803},{3,999872610,2999617830},{3,999897126,2999691378},{3,999897261,2999691783},{3,999946482,2999839446},{3,999964482,2999893446},{3,999970125,2999910375},{3,999970251,2999910753},{3,999971025,2999913075},{3,999971250,2999913750},{3,999972501,2999917503},{3,999972510,2999917530},{3,999987126,2999961378},{3,999987261,2999961783},{3,999997125,2999991375},{3,999997251,2999991753}}}

王守恩

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 A 的位数,  b 是 B 的位数。会有解吗？

northwolves

A370676
Number of unordered pairs of natural numbers k1, k2 such that their product is an n-digit number and has the same multiset of digits as in both k1 and k2.

0, 0, 3, 15, 98, 596, 3626, 22704, 146834, 983476, 6846451, 49364315, 367660050

COMMENTS
Since multiplication and multiset union are commutative operations, we count unordered pairs, i.e., we can assume that k1 <= k2.
The sequence could be redefined in terms of the number of distinct n-digit numbers that could be factorized into such pairs.
From David A. Corneth, Feb 27 2024: (Start)
a(n) >= 2*a(n-1).
As we need k1 + k2 == k1 * k2 (mod 9) there are two possible pairs of residues (k1, k2) mod 3, namely, (0, 0) and (2, 2), and six possible residues mod 9, namely, (0, 0), (2, 2), (3, 6), (5, 8), (6, 3), (8, 5). (End)
LINKS
Table of n, a(n) for n=1..13.
EXAMPLE
For n=3 the a(3)=3 solutions are:
  3 * 51 = 153
  6 * 21 = 126
  8 * 86 = 688
For n=4 the a(4)=15 solutions are:
  3 * 501 = 1503
  3 * 510 = 1530
  5 * 251 = 1255
  6 * 201 = 1206
  6 * 210 = 1260
  8 * 473 = 3784
  8 * 860 = 6880
  9 * 351 = 3159
  15 * 93 = 1395
  21 * 60 = 1260
  21 * 87 = 1827
  27 * 81 = 2187
  30 * 51 = 1530
  35 * 41 = 1435
  80 * 86 = 6880
PROG
(Python)

1. def a(n):
   
2.     count = 0
   
3.     for i in range(1, 10**(n-1)):
   
4.         for j in range(i, 10**n//i+1):
   
5.             if len(str(i*j)) == n and sorted(str(i)+str(j)) == sorted(str(i*j)):
   
6.                 count += 1
   
7.     print(n, count)

复制代码