将 3,4,…,1155 排成数列 {ak：k=1,2,…,1153}，使 ak 都是 k 的倍数，有几种排法？

王守恩

细化7#,10#。谢谢 lihp2020!  谢谢时空伴随者!
将 3,4,…,1155 排成数列 {ak: k=1,2,…,1153}, 使a(k)都是k的倍数, 有75种排法。
a(1)=3,a(3)=15,a(15)=105,a(105)=1155。
01:3,15,105,1155,
02:3,15,165,1155,
03:3,15,1155,
04:3,21,105,1155,
05:3,21,231,1155,
06:3,21,1155,
07:3,33,165,1155,
08:3,33,231,1155,
09:3,33,1155,
10:3,105,1155,
11:3,165,1155,
12:3,231,1155,
13:3,1155,
14:5,15,105,1155,
15:5,15,165,1155,
16:5,15,1155,
17:5,35,105,1155,
18:5,35,385,1155,
19:5,35,1155,
20:5,55,165,1155,
21:5,55,385,1155,
22:5,55,1155,
23:5,105,1155,
24:5,165,1155,
25:5,385,1155,
26:5,1155,
27:7,21,105,1155,
28:7,21,231,1155,
29:7,21,1155,
30:7,35,105,1155,
31:7,35,385,1155,
32:7,35,1155,
33:7,77,231,1155,
34:7,77,385,1155,
35:7,77,1155,
36:7,105,1155,
37:7,231,1155,
38:7,385,1155,
39:7,1155,
40:11,33,165,1155,
41:11,33,231,1155,
42:11,33,1155,
43:11,55,165,1155,
44:11,55,385,1155,
45:11,55,1155,
46:11,77,231,1155,
47:11,77,385,1155,
48:11,77,1155,
49:11,165,1155,
50:11,231,1155,
51:11,385,1155,
52:11,1155,
53:15,105,1155
54:15,165,1155
55:15,1155,
56:21,105,1155
57:21,231,1155
58:21,1155,
59:33,165,1155
60:33,231,1155
61:33,1155,
62:35,105,1155
63:35,385,1155
64:35,1155,
65:55,165,1155
66:55,385,1155
67:55,1155,
68:77,231,1155
69:77,385,1155
70:77,1155,
71:105,1155,
72:165,1155,
73:231,1155,
74:385,1155,
75:1155,

王守恩

加深影响。谢谢 lihp2020!  谢谢时空伴随者!
将 3,4,…,210 排成数列 {a(k): k=1,2,…,208}, 使a(k)都是k的倍数, 有39种排法。
a(1)=11,a(11)=209;a(2)=6,a(6)=30,a(30)=210。
01:11,209;6,30,210,
02:11,209;6,42,210,
03:11,209;6,210,
04:11,209;10,30,210,
05:11,209;10,70,210,
06:11,209;10,210,
07:11,209;14,42,210,
08:11,209;14,70,210,
09:11,209;14,210,
10:11,209;30,210,
11:11,209;42,210,
12:11,209;70,210,
13:11,209;210,
01:19,209;6,30,210,
02:19,209;6,42,210,
03:19,209;6,210,
04:19,209;10,30,210,
05:19,209;10,70,210,
06:19,209;10,210,
07:19,209;14,42,210,
08:19,209;14,70,210,
09:19,209;14,210,
10:19,209;30,210,
11:19,209;42,210,
12:19,209;70,210,
13:19,209;210,
01:209;6,30,210,
02:209;6,42,210,
03:209;6,210,
04:209;10,30,210,
05:209;10,70,210,
06:209;10,210,
07:209;14,42,210,
08:209;14,70,210,
09:209;14,210,
10:209;30,210,
11:209;42,210,
12:209;70,210,
13:209;210,

王守恩

将3,4,…,n排成数列{a(k):k=1,2,3,4,5,…,n-2},使a(k)都是k的倍数,有几种排法?
a(02)=b(03)*b(04)=1*1=1,{3,4},
a(03)=b(04)*b(05)=1*1=1,{5,4,3},
a(04)=b(05)*b(06)=1*1=1,{5,6,3,4},
a(05)=b(06)*b(07)=1*1=1,{7,6,3,4,5},
a(06)=b(07)*b(08)=1*2=2,{7,4,3,8,5,6},{7,8,3,4,5,6},
a(07)=b(08)*b(09)=2*2=4,{3,4,8,9,5,6,7},{3,8,9,4,5,6,7},{9,4,3,8,5,6,7}{9,8,3,4,5,6,7},
a(08)=b(09)*b(10)=2*1=2,{3,10,9,4,5,6,7,8},{9,10,3,4,5,6,7,8},
a(09)=b(10)*b(11)=1*1=1,{11,10,3,4,5,6,7,8,9},
a(10)=b(11)*b(12)=1*3=3,{11,4,3,12,5,6,7,8,9,10},{11,6,3,4,5,12,7,8,9,10},{11,12,3,4,5,6,7,8,9,10},
a(11)=b(12)*b(13)=3*1=3,
a(12)=b(13)*b(14)=1*1=1,
a(13)=b(14)*b(15)=1*3=3,
其中:
b(03)=1,{3},   (奇数约数, 除了"1")
b(04)=1,{4},   (偶数约数, 除了"2")
b(05)=1,{5},   (奇数约数, 除了"1")
b(06)=1,{6},   (偶数约数, 除了"2")
b(07)=1,{7},          .........
b(08)=2,{4,8},{8},
b(09)=2,{3,9},{9},
b(10)=1,{10},
b(11)=1,{11},
b(12)=3,{4,12},{6,12},{12},
b(13)=1,{13},
b(14)=1,{14},
b(15)=3,{3,15},{5,15},{15},
b(16)=4,{4,8,16},{4,16},{8,16},{16}
b(17)=1,{17},
b(18)=2,{6,18},{18},
b(19)=1,{19},
b(20)=3,{4,20},{10,20},{20},
b(21)=3,{3,21},{7,21},{21},
b(22)=1,{22},
b(23)=1,{23},
b(24)=8,{4,8,24},{4,12,24},{4,24},{6,12,24},{6,24},{8,24},{12,24},{24},
b(25)=2,{5,25},{25},
b(26)=1,{26},
b(27)=4,{3,9,27},{3,27},{9,27},{27},
b(28)=3,{4,28},{14,28},{28},
b(29)=1,{29},
b(30)=3,{6,30},{10,30},{30},
b(31)=1,{31},
b(32)=8,{4,8,16,32},{4,8,32},{4,16,32},{4,32},{8,16,32},{8,32},{16,32},{32},
b(33)=3,{3,33},{11,33},{33},
b(n)可以有通项公式吗？谢谢各位！

王守恩

再归纳，各位可有好办法？
b(03)=1,{3},   
b(04)=1,{4},   
b(05)=1,{5},   
b(06)=1,{6},   
b(07)=1,{7},         
b(08)=2,{4,8},{8},
b(09)=2,{3,9},{9},
b(10)=1,{10},
b(11)=1,{11},
b(12)=3,{4,12},{6,12},{12},
b(13)=1,{13},
b(14)=1,{14},
b(15)=3,{3,15},{5,15},{15},
b(16)=4,{4,8,16},{4,16},{8,16},{16}
b(17)=1,{17},
b(18)=2,{6,18},{18},
b(19)=1,{19},
b(20)=3,{4,20},{10,20},{20},
......
01=3,5,07,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,
01=4,6,10,14,22,26,34,38,46,58,62,74,82,86,94,
02=9,25,49,121,
02=8,18,50,098,
03=15,21,33,35,39,51,55,57,65,69,77,85,87,93,95,
03=12,20,28,30,42,44,52,66,68,70,76,78,92,
04=27,125,
04=16,054,
08=45,63,75,81,99,
08=24,32,36,40,56,88,
09=90,
13=60,
16=64,
20=48,080,
26=72,200,

王守恩

将11#归纳一下。
              1155,385,231,165,105,77,55,35,33,21,15,
1155=01  1
0385=01  1
0231=01  1
0165=01  1
0105=01  1
0077=03  1     1     1
0055=03  1     1            1
0035=03  1     1                   1
0033=03  1            1     1
0021=03  1            1            1
0015=03  1                   1     1
0011=13  1     1     1     1           3   3        3
0007=13  1     1     1            1    3        3       3
0005=13  1     1            1     1         3   3            3
0003=13  1            1     1     1                   3  3   3
              1155,385,231,165,105,77,55,35,33,21,15,

合计：1(1155)+1(385)+1(231)+1(165)+1(105)+3(77)+3(55)+3(35)
+3(33)+3(21)+3(15)+13(11)+13(7)+13(5)+13(3)=75
用这方法做具体的题目没问题，关键是怎么总结出个通项公式来。

allen125

謝謝各位

cgl_74

这个题，求通项公式比较复杂。但是在特定情况下可以求出递推公式，解决指定具体值的排列数，也是很快的！



注：还有个限制，即偶数排列是唯一的。也即x-1=2*质数的情况。本例1155-1=1154=2*577

cgl_74

补充一个稍微完整点的递推公式解法，把偶数排法也考虑进去：

王守恩

将3,4,5,6,…,n排成数列{a(k):k=1,2,3,4,5,…,n-2},使a(k)都是k的倍数,有几种排法?
a(03)=b(01)*b(03)=1,{3},
a(04)=b(03)*b(02)=1,{3,4},
a(05)=b(02)*b(05)=1,{5,4,3},
a(06)=b(05)*b(03)=1,{5,6,3,4},
a(07)=b(03)*b(07)=1,{7,6,3,4,5},
a(08)=b(07)*b(04)=2,{7,4,3,8,5,6},{7,8,3,4,5,6},
a(09)=b(04)*b(09)=4,{3,4,8,9,5,6,7},{3,8,9,4,5,6,7},{9,4,3,8,5,6,7}{9,8,3,4,5,6,7},
a(10)=b(09)*b(05)=2,{3,10,9,4,5,6,7,8},{9,10,3,4,5,6,7,8},
a(11)=b(05)*b(11)=1,{11,10,3,4,5,6,7,8,9},
a(12)=b(11)*b(06)=3,{11,4,3,12,5,6,7,8,9,10},{11,6,3,4,5,12,7,8,9,10},{11,12,3,4,5,6,7,8,9,10},
a(13)=b(06)*b(13)=3,{13,4,3,12,5,6,7,8,9,10,11},{13,6,3,4,5,12,7,8,9,10,11},{13,12,3,4,5,6,7,8,9,10,11},
a(14)=b(13)*b(07)=1,{13,14,3,4,5,6,7,8,9,10,11,12},
......
Table[Total[1/Divisors[n]]; {b[Floor[n/2]]*b[2 Floor[n/2] - Cos[n*Pi]]}, {n,3, 210}]
{1, 1, 1, 1, 1, 2, 4, 2, 1, 3, 3, 1, 3, 12, 4, 2, 2, 3, 9, 3, 1, 8, 16, 2, 4, 12, 3, 3, 3, 8, 24, 3, 3, 24, 8,
1, 3, 24, 8, 3, 3, 3, 24, 8, 1, 20, 40, 4, 6, 9, 3, 4, 12, 24, 24, 3, 1, 13, 13, 1, 8, 128, 48, 9, 3, 3,
9, 9, 3, 26, 26, 1, 8, 24, 9, 9, 3, 20, 160, 8, 1, 13, 39, 3, 3, 24, 8, 8, 24, 9, 9, 3, 3, 144, 48, 2, 16,
64, 8, 3, 3, 8, 104, 13, 1, 20, 20, 3, 9, 60, 20, 3, 9, 9, 24, 8, 3, 132, 88, 2, 3, 9, 12, 32, 8, 32, 96,
9, 3, 13, 39, 3, 20,160, 8, 3, 3, 13, 39, 3, 3, 228, 228, 3, 8, 24, 3, 8, 8, 8, 64, 24, 9, 39, 13, 1, 3,
144, 144, 24, 8, 3, 39, 13, 1, 44, 88, 6, 24, 24, 3, 3, 24, 160, 60, 3, 1, 44, 44, 3, 9, 24, 24, 9, 9,
9, 60, 60, 3, 112, 112, 1, 13, 104, 8, 8, 8, 26, 78, 3, 3, 39, 39, 3, 8, 160, 60, 39}
譬如:
a(210)=b(210/2)*b(209)=b(3*5*7)*b(11*19)=13*3=39,
a(1155)=b((1155-1)/2)*b(1155)=b(577)*b(3*5*7*11)=1*75=75,
a(3*5*5*7*7*7*11*11)=b((3*5*5*7*7*7*11*11 - 1)/2)*b(3*5*5*7*7*7*11*11)
=b(2351*331*2/2)*b(3*5*5*7*7*7*11*11)=13*34516=448708,