3^n倍数的n位数

王守恩

A(n)=十进制的n位数,  每个数位只由数码 2,3,4,5 构成,   A(n)是3^n的倍数。譬如
A(1)=3,
A(2)=45,
A(3)=243,
A(4)=4455,
A(5)=24543,
A(6)=222345, ——有多个解时取最小的那个。
A(7)=3444525,
A(8)=23324355,
A(9)=225252252,

Table[SelectFirst[FromDigits /@ Tuples[{2, 3, 4, 5}, n], Mod[#, 3^n] == 0 &], {n, 9}]

我只会用这个。大了就来不了。各位大侠！再来几个开开眼。谢谢！！！

王守恩

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个不同数码构成, 可以有74个组合。——明显无解的已删除。——不知这74个组合是否都"有解"？——疑惑？
{1236}, {1239}, {1249}, {1259}, {1267}, {1268}, {1269}, {1279}, {1289}, {1345},
{1349}, {1356}, {1358}, {1359}, {1369}, {1378}, {1379}, {1389}, {1459}, {1468},
{2378}, {1469}, {1479}, {1489}, {1569}, {1579}, {1589}, {1679}, {1689}, {1789},
{2345}, {2346}, {2347}, {2349}, {2359}, {2369}, {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}, {3469}, {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}——这串数可以无限长叫"有解"。
A{1378}={3, 18, 378, 1377, 11178, 137781, 1113183, 11173383, 137111778, 1171118817, 13177788183, 111173737113}
A{2378}={3, 27, 378, 3888, 23328, 238383, 2283228, 27838323, 222732828, 2327888727, 22722822837, 222783787287}——再长我就来不了了——各位大侠！再长几个开开眼。谢谢！！！
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}——OEIS可是没有这些数字串的。

补充内容 (2025-9-17 14:27):
同上,9不能有,可以有18个组合。——去掉3469,  增加3678。

northwolves

1. f[n_]:=(a=3^n;For[r=Ceiling[2*(10^n-1)/(9a)],r<=Floor[5*(10^n-1)/(9a)],r++,k=r*a;If[SubsetQ[{2,3,4,5},Union[IntegerDigits@k]],Return["A("<>ToString@n<>")="<>ToString[k]]]]);
   
2. Do[Print[f[n]],{n,18}]

复制代码

A(1)=3
A(2)=45
A(3)=243
A(4)=4455
A(5)=24543
A(6)=222345
A(7)=3444525
A(8)=23324355
A(9)=225252252
A(10)=2432523555
A(11)=22223445444
A(12)=225242233353
A(13)=2232345555432
A(14)=22235525452224
A(15)=224552445255522
A(16)=2225242344255255
A(17)=22223225454352344
A(18)=222345533453423454

northwolves

王守恩 发表于 2025-9-16 11:34
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个不同数码构成, 可以有74个组合 ...

1. f[c_,n_]:=(a=3^n;b=(10^n-1)/(9a);For[r=Ceiling[c[[1]]*b],r<=Floor[c[[4]]*b],r++,k=r*a;
   
2. If[SubsetQ[c,Union[IntegerDigits@k]],Return[k]]]);
   
3. g[c_]:=Table[f[IntegerDigits@c,n],{n,20}];
   
4. g@2378

复制代码

{3,27,378,3888,23328,238383,2283228,27838323,222732828,2327888727,22722822837,222783787287,2227828838373,22723828323372,222733333872783,2222232388382772,22323337233232878,222338738872887372,2223373333888373778,22223327328377888832}

王守恩

9个1位数{1, 2, 3, 4, 5, 6, 7, 8, 9}地位是相等的。一般的题目, 能把这9个数搞清楚, 也就差不离了。譬如

a(n) contains n digits (either '2' or '3') and is divisible by 2^n.

A(2, 1) = {2, 12, 112, 2112, 22112, 122112, 2122112, 12122112, 212122112, 1212122112}, ——A053312——2025。
A(2, 3) = {2, 32, 232, 3232, 23232, 223232, 2223232, 32223232, 232223232, 3232223232}, ——A053316——2019。
A(2, 5) = {2, 52, 552, 5552, 55552, 255552, 5255552, 55255552, 255255552, 2255255552}, ——A053317——2022。
A(2, 7) = {2, 72, 272, 2272, 22272, 222272, 7222272, 27222272, 727222272, 2727222272}, ——A053318——2025。
A(2, 9) = {2, 92, 992, 2992, 92992, 292992, 2292992, 22292992, 222292992, 2222292992}, ——A053313——2015。
A(4, 1) = {4, 44, 144, 4144, 14144, 414144, 1414144, 41414144, 441414144, 1441414144}, ——A053314——2022。
A(4, 3) = {4, 44, 344, 3344, 33344, 433344, 3433344, 33433344, 333433344, 3333433344}, ——A035014——2017。
A(4, 5) = {4, 44, 544, 4544, 44544, 444544, 4444544, 54444544, 454444544, 5454444544}, ——A053315——2019。
A(4, 7) = {4, 44, 744, 7744, 47744, 447744, 4447744, 44447744, 444447744, 4444447744}, ——A053332——2017。
A(4, 9) = {4, 44, 944, 4944, 94944, 994944, 4994944, 94994944, 494994944, 9494994944}, ——A053333——2023。
A(6, 1) = {6, 16, 616, 1616, 11616, 111616, 6111616, 16111616, 616111616, 1616111616}, ——A053334——2025。
A(6, 3) = {6, 36, 336, 6336, 66336, 366336, 6366336, 36366336, 636366336, 3636366336}, ——A053335——2000。
A(6, 5) = {6, 56, 656, 6656, 66656, 566656, 6566656, 66566656, 666566656, 6666566656}, ——A053336——2020。
A(6, 7) = {6, 76, 776, 7776, 67776, 667776, 6667776, 66667776, 766667776, 6766667776}, ——A053337——2016。
A(6, 9) = {6, 96, 696, 9696, 69696, 669696, 6669696, 96669696, 696669696, 9696669696}, ——A053338——2023。
A(8, 1) = {8, 88, 888, 1888, 81888, 181888, 8181888, 18181888, 118181888, 8118181888}, ——A053376——2017。
A(8, 3) = {8, 88, 888, 3888, 33888, 333888, 3333888, 83333888, 383333888, 3383333888}, ——A053377——2015。
A(8, 5) = {8, 88, 888, 5888, 85888, 885888, 8885888, 58885888, 558885888, 8558885888}, ——A053378——2000。
A(8, 7) = {8, 88, 888, 7888, 77888, 877888, 7877888, 87877888, 787877888, 8787877888}, ——A053379——2000。
A(8, 9) = {8, 88, 888, 9888, 89888, 989888, 9989888, 89989888, 989989888, 8989989888}, ——A053380——2000。

20个数字串。差距是在通项公式。越后越好！

Table[SelectFirst[FromDigits /@ Tuples[{5, 8}, n], Mod[#, 2^n] == 0 &], {n, 10}]

nxt[{n_, a_}] := {n + 1, a + 10^n (6 - 5*Mod[a/2^n, 2])}; NestList[nxt, {1, 6}, 20][[;; , 2]]

mathe

限制长度没有必要，可以求最小的整数a(n)为3^n的倍数而且仅由数字1和2构成，比如a(1)=12,a(2)=12222

mathe

3:21222
4:21222
5:1112211
6:21111111
7:21111111
8:12221122212
9:211212112221
10:12211212222121221
11:22122212212121121
12:111122221211112222
13:111122221211112222
14:12121112111211112212
15:22111121212221122212221
16:22111121212221122212221
17:1111211221212221122211121
18:111212121221122121222112122121
19:112122112212212222221211111112
20:112122112212212222221211111112
21:21122122221211121211121112111112
22:122122221211222222222122212221122222
23:122122221211222222222122212221122222
24:21111212122212122212111111222211112111
25:21111212122212122212111111222211112111
26:1222111212222212221212221121221211122121
27:11112222212122112121122111111112211221111222
28:11112222212122112121122111111112211221111222
29:11112222212122112121122111111112211221111222
30:11212112122122212122222221221222212221121212
31:11212112122122212122222221221222212221121212

王守恩

再开放一点。3^n倍数的n位数——n位数=3个数码。—— 3,6,9 是最有诱惑力的——还有吗？

A(n)=十进制的n位数。A(n)是3^n的倍数。 每个数位只由数码 3,6,9 构成。譬如
A(1)=3,
A(2)=36,
A(3)=999,
A(4)=3969,
A(5)=36693,
A(6)=639333,
A(7)=6969969,
A(8)=63333333,
A(9)=339393969,
A(10)=6636339963,
A(11)=33333396696,
A(12)=963969669639,
A(13)=3393696663999,
A(14)=33969669393366,
A(15)=好像没有？
......

后面的都会有吗？！？！

我这里来不了。

northwolves

A(16)=9339933966699699
A(17)=36339999639366699

mathe

3:999
4:3969
5:36693
6:639333
7:6969969
8:63333333
9:339393969
10:6636339963
11:33333396696
12:963969669639
13:3393696663999
14:33969669393366
15:-1
16:9339933966699699
17:36339999639366699
18:-1
19:3393639369996336666
20:39933663936639393366
21:393693699699633693969
22:3399336339636366339336
23:33396999363963396993636
24:333339333639669939966399
25:3639333969393369636666366
26:69333936339369393363366963
27:-1
28:6399699669366993336339663633
29:-1
30:333396939369633936396396366663
31:3633339336666336963666339369939
32:39333396366696969366639369693966