3^n倍数的n位数

王守恩

小结。——2#的瑕疵已剔除。
A(n)=十进制的n位数, 且A(n)是3^n的倍数。 每个数位只由9个数码(1—9)中的4个(3个无解)不同数码构成, 可以有74个组合(不能多也不能少)。——74个数字串——每个可以无限长——OEIS没有这74个数字串。
{1236}, {1239}, {1249}, {1259}, {1267}, {1268}, {1269}, {1279}, {1289}, {1345},
{1349}, {1356}, {1358}, {1359}, {1369}, {1378}, {1379}, {1389}, {1459}, {1468},
{1469}, {1479}, {1489}, {1569}, {1579}, {1589}, {1679}, {1689}, {1789}, {2345},
{2346}, {2347}, {2349}, {2359}, {2369}, {2378}, {2379}, {2389}, {2459}, {2469},
{2479}, {2489}, {2567}, {2569}, {2579}, {2589}, {2679}, {2689}, {2789}, {3459},
{3468}, {3469}, {3479}, {3489}, {3567}, {3569}, {3579}, {3589}, {3678}, {3679},
{3689}, {3789}, {4567}, {4568}, {4569}, {4579}, {4589}, {4679}, {4689}, {4789},
{5679}, {5689}, {5789}, {6789},

同上,   9不能有,   可以有18个组合。
{1236}, {1267}, {1268}, {1345}, {1356}, {1358},
{1378}, {1468}, {2378}, {2345}, {2346}, {2347},
{2567}, {3468}, {3678}, {3567}, {4567}, {4568},

同上,   9与6都不能有,   只有6个组合。
A{1345}={3, 45, 135, 1134, 33534, 114453, 1454355, 11114334, 113531544, 1133445555, 11533155435, 113315444343, ......}
A{1358}={3, 18, 135, 3888, 13851, 188811, 1113183, 11383335, 113885838, 1355883138, 11538115551, 111835381158, ......}
A{1378}={3, 18, 378, 1377, 11178, 137781, 1113183, 11173383, 137111778, 1171118817, 13177788183, 111173737113, ......}——OEIS没有这些数字串。
A{2378}={3, 27, 378, 3888, 23328, 238383, 2283228, 27838323, 222732828, 2327888727, 22722822837, 222783787287, ......}
A{2345}={3, 45, 243, 4455, 24543, 222345, 3444525, 23324355, 225252252, 2432523555, 22223445444, 225242233353, ......}
A{2347}={3, 27, 243, 4374, 33777, 332424, 2342277, 23337477, 322427223, 2344422447, 22477474242, 244224774432, ......}——这些数字串,比5#的"困难"多了。

northwolves

northwolves 发表于 2025-9-16 20:55
A(1)=3
A(2)=45
A(3)=243

1. f[n_] := Module[{a, lower, upper, valid}, a = 3^n;
   
2.   lower = Ceiling[2*(10^n - 1)/(9 a)];  upper = Floor[5*(10^n - 1)/(9 a)];
   
3.   valid = Select[a*Range[lower, upper],  SubsetQ[{2, 3, 4, 5}, Union[IntegerDigits[#]]] &];
   
4.   {n, Length[valid], valid}]
   
5. Do[Print[f[n]], {n, 15}]

复制代码

不是递增的：
{1,1,{3}}
{2,2,{45,54}}
{3,3,{243,324,432}}
{4,1,{4455}}
{5,7,{24543,33534,35235,42525,43254,52245,54432}}
{6,6,{222345,224532,252234,332424,443232,523422}}
{7,5,{3444525,3525444,4553334,5325345,5443443}}
{8,9,{23324355,23442453,24354432,35252253,35324424,35442522,43532235,52245243,55322352}}
{9,7,{225252252,245545425,323352324,435545424,455425254,455543352,524453535}}
{10,24,{2432523555,2444333355,2545425243,2554223544,3325344435,3425255343,3443442435,3455252235,3543235245,3543353343,3544534323,3554454555,4242434454,4242552552,4254244254,4255425234,4255543332,4542344325,5253353334,5354445222,5423532552,5435224254,5435342352,5553322254}}
{11,26,{22223445444,22245234525,22354534224,23222554524,23534333244,24233532453,24355232442,24533442324,25244333235,25343535555,32353242345,32442524433,34252435332,35332323444,35345432322,35355352554,35432234352,43343442225,43423335522,43542555453,45252555444,52432323354,52445432232,53253222552,53555435334,54242234253}}
{12,40,{225242233353,233352554454,242424252324,244545233355,245522553354,245544342435,254222242524,255535433235,322332252525,325552253544,332323343325,332423254233,335325453534,335543344344,343254553254,344545423443,345544532523,353223323532,422322345234,424225435455,424553334552,425224544535,425324455443,425442435345,434544425352,435332552355,435454252344,435543534432,454522355424,522233253234,522322535322,525353343345,525453254253,532452332223,534355422444,534455333352,535454442432,545445533232,552254355324,553423525524}}
{13,55,{2232345555432,2233525354452,2323424445453,2323542425355,2324545254522,2344544442234,2353222342323,2423235442545,2424552353343,2434325553333,2434443533235,2443545523242,2542532255343,2552423435235,2552533443522,2555425545444,3232534523544,3242355553224,3243535352244,3253254345252,3255333342444,3333434443245,3345232433445,3354334423452,3355345224234,3423443542533,3432435524253,3445354323522,3523455424323,4232425353255,4244223343455,4322434452543,4322552432445,4323445253325,4335243243525,4344235225245,4344345233532,4355553324222,4434244324533,4444352332353,4522335453252,4554435552534,5222525445423,5245442244225,5254544234232,5322532544244,5333235234543,5335223355324,5344443325233,5353234422255,5444254322325,5533552353555,5535352344222,5543325553545,5552545523454}}
{14,45,{22235525452224,22245244445232,22253423322222,22542544232334,23543232223545,24243554544525,24455253535434,25234442232255,25324223343354,25523534444553,25524352332252,25545445225542,32232222423333,32442534353232,32535553534344,33225353323524,33443222344443,34224233352453,34225333435323,34324245234243,34453232342235,35225452253223,35242532235522,42343232353542,42433525242324,42443244235332,42553252522332,43224255243342,43255234533555,43345355235453,43552553452533,44344522242522,45223244524233,45242223345225,52452453453345,52534242223245,53225443422342,53244422243334,53342243525322,53544223523223,54244354525443,54423333225423,54543524453424,55325444225544,55334254454442}}
{15,70,{224552445255522,225443354542245,232422433324533,234243553554252,234424335433545,242453553323535,243224534445552,243543453252534,245235232434555,245352233422233,245542442533425,254235254325444,254442452542524,322255243535454,323424435524535,323453334223233,324242323223535,324424554342435,332353344232332,332543553343524,333552324552345,334235332525545,334255435344252,335325232454544,335442233442222,343252343522322,344325254345433,344333433222423,345554324323425,352223323225443,354225225334455,354354523335432,354454334332524,354532435433325,423255535435242,423544522422222,424325332545534,432245355253254,432342253422225,432533524352535,434223452525553,435335435422425,442555432353243,443225325425445,443444433235335,444225444243345,445542444323433,452344443244434,452422544345235,452434224355533,453242254355424,454343432525325,454442454332532,454522234255452,455433332454324,525254353424253,525542522523534,532453343555223,533225544334335,533242332555525,533534433255324,535323555422433,535325234244552,543432553235343,543442554423522,544553432454555,545252224225455,545434455344355,553523522525442,553553454345444}}

王守恩

northwolves 发表于 2025-9-19 13:46
不是递增的：
{1,1,{3}}
{2,2,{45,54}}

谢谢！！！不是递增的。——应在意料之中。这些数字串我这里拉不长。——5#的数字串我能拉得长长的。

目标很明确: 74个组合要一起”无限长“。只要有1个不能”无限长“, 74个组合就全废,这个帖子就丢了。不可惜！

74个组合*4=296个数码=56(数码9)+33*2(数码3,6)+29*6(数码1,2,4,5,7,8)。——机会是均等的。