将 1、2、3、4、5、6、7 排成一列，使得相邻两数之和为质数，有几种排列方式？

wintex

請問數學

天山草

这题是啥意思？你先给出其中的一个排列，让大家看看是啥意思。

wintex

天山草 发表于 2025-4-29 09:45
这题是啥意思？你先给出其中的一个排列，让大家看看是啥意思。

1,2,3,4 ,四個數，相鄰數相加都是質數
1,2,4,3 不行，2+4=6

luyuanhong

   1、2、3、4、7、6、5 就是一个符合条件的排列。因为

1+2=3，2+3=5，3+4=7，7+6=13，6+5=11 都是质数。

时空伴随者

奇偶相间排列
2025-04-29 13:46:52
1:  1 4 3 2 5 6 7
2:  1 2 3 4 7 6 5
3:  1 6 5 2 3 4 7
4:  1 2 5 6 7 4 3
5:  1 6 7 4 3 2 5
6:  1 4 7 6 5 2 3
7:  3 4 1 2 5 6 7
8:  3 2 1 4 7 6 5
9:  3 2 5 6 1 4 7
10:  3 2 5 6 7 4 1
11:  3 4 7 6 1 2 5
12:  3 4 7 6 5 2 1
13:  5 6 1 2 3 4 7
14:  5 2 1 6 7 4 3
15:  5 2 3 4 1 6 7
16:  5 2 3 4 7 6 1
17:  5 6 7 4 1 2 3
18:  5 6 7 4 3 2 1
19:  7 6 1 4 3 2 5
20:  7 4 1 6 5 2 3
21:  7 4 3 2 1 6 5
22:  7 4 3 2 5 6 1
23:  7 6 5 2 1 4 3
24:  7 6 5 2 3 4 1

天山草

写个简单程序硬算：

1. Clear["Global`*"];
   
2. a = Permutations[{1, 2, 3, 4, 5, 6, 7}];(*1,2,3,4,5,6,7 全排列*)
   
3. n = 0;
   
4. Do[c = a[[k]];
   
5. If[PrimeQ [c[[1]] + c[[2]]] && PrimeQ [c[[2]] + c[[3]]] &&
   
6.    PrimeQ [c[[3]] + c[[4]]] && PrimeQ [c[[4]] + c[[5]]] &&
   
7.    PrimeQ [c[[5]] + c[[6]]] && PrimeQ [c[[6]] + c[[7]]], n = n + 1;
   
8.   Print[n, "----", c]], {k, 1, 7!}]

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运行结果：

1. 1----{1,2,3,4,7,6,5}
   
2. 2----{1,2,5,6,7,4,3}
   
3. 3----{1,4,3,2,5,6,7}
   
4. 4----{1,4,7,6,5,2,3}
   
5. 5----{1,6,5,2,3,4,7}
   
6. 6----{1,6,7,4,3,2,5}
   
7. 7----{3,2,1,4,7,6,5}
   
8. 8----{3,2,5,6,1,4,7}
   
9. 9----{3,2,5,6,7,4,1}
   
10. 10----{3,4,1,2,5,6,7}
    
11. 11----{3,4,7,6,1,2,5}
    
12. 12----{3,4,7,6,5,2,1}
    
13. 13----{5,2,1,6,7,4,3}
    
14. 14----{5,2,3,4,1,6,7}
    
15. 15----{5,2,3,4,7,6,1}
    
16. 16----{5,6,1,2,3,4,7}
    
17. 17----{5,6,7,4,1,2,3}
    
18. 18----{5,6,7,4,3,2,1}
    
19. 19----{7,4,1,6,5,2,3}
    
20. 20----{7,4,3,2,1,6,5}
    
21. 21----{7,4,3,2,5,6,1}
    
22. 22----{7,6,1,4,3,2,5}
    
23. 23----{7,6,5,2,1,4,3}
    
24. 24----{7,6,5,2,3,4,1}

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七个数字共有 24 种排列方案。

天山草

1，2，3，4，5，6，7，8  八个数字有 60 种排列方案：

1. 1----{1,2,3,4,7,6,5,8}
   
2. 2----{1,2,3,8,5,6,7,4}
   
3. 3----{1,2,5,6,7,4,3,8}
   
4. 4----{1,2,5,8,3,4,7,6}
   
5. 5----{1,4,7,6,5,2,3,8}
   
6. 6----{1,4,7,6,5,8,3,2}
   
7. 7----{1,6,7,4,3,2,5,8}
   
8. 8----{1,6,7,4,3,8,5,2}
   
9. 9----{2,1,4,3,8,5,6,7}
   
10. 10----{2,1,4,7,6,5,8,3}
    
11. 11----{2,1,6,5,8,3,4,7}
    
12. 12----{2,1,6,7,4,3,8,5}
    
13. 13----{2,3,8,5,6,1,4,7}
    
14. 14----{2,3,8,5,6,7,4,1}
    
15. 15----{2,5,8,3,4,1,6,7}
    
16. 16----{2,5,8,3,4,7,6,1}
    
17. 17----{3,2,1,4,7,6,5,8}
    
18. 18----{3,4,7,6,1,2,5,8}
    
19. 19----{3,8,5,2,1,4,7,6}
    
20. 20----{3,8,5,2,1,6,7,4}
    
21. 21----{3,8,5,6,7,4,1,2}
    
22. 22----{4,1,2,3,8,5,6,7}
    
23. 23----{4,3,8,5,2,1,6,7}
    
24. 24----{4,7,6,1,2,3,8,5}
    
25. 25----{4,7,6,1,2,5,8,3}
    
26. 26----{4,7,6,5,8,3,2,1}
    
27. 27----{5,2,1,6,7,4,3,8}
    
28. 28----{5,6,7,4,1,2,3,8}
    
29. 29----{5,8,3,2,1,4,7,6}
    
30. 30----{5,8,3,2,1,6,7,4}
    
31. 31----{5,8,3,4,7,6,1,2}
    
32. 32----{6,1,2,5,8,3,4,7}
    
33. 33----{6,5,8,3,2,1,4,7}
    
34. 34----{6,7,4,1,2,3,8,5}
    
35. 35----{6,7,4,1,2,5,8,3}
    
36. 36----{6,7,4,3,8,5,2,1}
    
37. 37----{7,4,1,2,3,8,5,6}
    
38. 38----{7,4,1,6,5,2,3,8}
    
39. 39----{7,4,1,6,5,8,3,2}
    
40. 40----{7,4,3,2,1,6,5,8}
    
41. 41----{7,4,3,8,5,2,1,6}
    
42. 42----{7,4,3,8,5,6,1,2}
    
43. 43----{7,6,1,2,5,8,3,4}
    
44. 44----{7,6,1,4,3,2,5,8}
    
45. 45----{7,6,1,4,3,8,5,2}
    
46. 46----{7,6,5,2,1,4,3,8}
    
47. 47----{7,6,5,8,3,2,1,4}
    
48. 48----{7,6,5,8,3,4,1,2}
    
49. 49----{8,3,2,1,4,7,6,5}
    
50. 50----{8,3,2,5,6,1,4,7}
    
51. 51----{8,3,2,5,6,7,4,1}
    
52. 52----{8,3,4,1,2,5,6,7}
    
53. 53----{8,3,4,7,6,1,2,5}
    
54. 54----{8,3,4,7,6,5,2,1}
    
55. 55----{8,5,2,1,6,7,4,3}
    
56. 56----{8,5,2,3,4,1,6,7}
    
57. 57----{8,5,2,3,4,7,6,1}
    
58. 58----{8,5,6,1,2,3,4,7}
    
59. 59----{8,5,6,7,4,1,2,3}
    
60. 60----{8,5,6,7,4,3,2,1}

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1，2，3，4，5，6，7，8，9 九个数字有 140 种排列方案：

1. 1----{1,2,3,4,7,6,5,8,9}
   
2. 2----{1,2,3,4,9,8,5,6,7}
   
3. 3----{1,2,3,8,5,6,7,4,9}
   
4. 4----{1,2,3,8,9,4,7,6,5}
   
5. 5----{1,2,5,6,7,4,3,8,9}
   
6. 6----{1,2,5,6,7,4,9,8,3}
   
7. 7----{1,2,9,4,3,8,5,6,7}
   
8. 8----{1,2,9,4,7,6,5,8,3}
   
9. 9----{1,2,9,8,3,4,7,6,5}
   
10. 10----{1,2,9,8,5,6,7,4,3}
    
11. 11----{1,4,3,2,9,8,5,6,7}
    
12. 12----{1,4,3,8,9,2,5,6,7}
    
13. 13----{1,4,7,6,5,2,3,8,9}
    
14. 14----{1,4,7,6,5,2,9,8,3}
    
15. 15----{1,4,7,6,5,8,3,2,9}
    
16. 16----{1,4,7,6,5,8,9,2,3}
    
17. 17----{1,4,9,2,3,8,5,6,7}
    
18. 18----{1,4,9,8,3,2,5,6,7}
    
19. 19----{1,6,5,2,3,8,9,4,7}
    
20. 20----{1,6,5,2,9,8,3,4,7}
    
21. 21----{1,6,5,8,3,2,9,4,7}
    
22. 22----{1,6,5,8,9,2,3,4,7}
    
23. 23----{1,6,7,4,3,2,5,8,9}
    
24. 24----{1,6,7,4,3,2,9,8,5}
    
25. 25----{1,6,7,4,3,8,5,2,9}
    
26. 26----{1,6,7,4,3,8,9,2,5}
    
27. 27----{1,6,7,4,9,2,3,8,5}
    
28. 28----{1,6,7,4,9,2,5,8,3}
    
29. 29----{1,6,7,4,9,8,3,2,5}
    
30. 30----{1,6,7,4,9,8,5,2,3}
    
31. 31----{3,2,1,4,7,6,5,8,9}
    
32. 32----{3,2,1,4,9,8,5,6,7}
    
33. 33----{3,2,1,6,5,8,9,4,7}
    
34. 34----{3,2,1,6,7,4,9,8,5}
    
35. 35----{3,2,5,8,9,4,1,6,7}
    
36. 36----{3,2,5,8,9,4,7,6,1}
    
37. 37----{3,2,9,8,5,6,1,4,7}
    
38. 38----{3,2,9,8,5,6,7,4,1}
    
39. 39----{3,4,1,2,9,8,5,6,7}
    
40. 40----{3,4,7,6,1,2,5,8,9}
    
41. 41----{3,4,7,6,1,2,9,8,5}
    
42. 42----{3,4,7,6,5,8,9,2,1}
    
43. 43----{3,4,9,8,5,2,1,6,7}
    
44. 44----{3,8,5,2,1,6,7,4,9}
    
45. 45----{3,8,5,2,9,4,1,6,7}
    
46. 46----{3,8,5,2,9,4,7,6,1}
    
47. 47----{3,8,5,6,1,2,9,4,7}
    
48. 48----{3,8,5,6,7,4,1,2,9}
    
49. 49----{3,8,5,6,7,4,9,2,1}
    
50. 50----{3,8,9,2,1,4,7,6,5}
    
51. 51----{3,8,9,2,5,6,1,4,7}
    
52. 52----{3,8,9,2,5,6,7,4,1}
    
53. 53----{3,8,9,4,1,2,5,6,7}
    
54. 54----{3,8,9,4,7,6,1,2,5}
    
55. 55----{3,8,9,4,7,6,5,2,1}
    
56. 56----{5,2,1,6,7,4,3,8,9}
    
57. 57----{5,2,1,6,7,4,9,8,3}
    
58. 58----{5,2,3,8,9,4,1,6,7}
    
59. 59----{5,2,3,8,9,4,7,6,1}
    
60. 60----{5,2,9,8,3,4,1,6,7}
    
61. 61----{5,2,9,8,3,4,7,6,1}
    
62. 62----{5,6,1,2,3,8,9,4,7}
    
63. 63----{5,6,1,2,9,8,3,4,7}
    
64. 64----{5,6,7,4,1,2,3,8,9}
    
65. 65----{5,6,7,4,1,2,9,8,3}
    
66. 66----{5,6,7,4,3,8,9,2,1}
    
67. 67----{5,6,7,4,9,8,3,2,1}
    
68. 68----{5,8,3,2,1,6,7,4,9}
    
69. 69----{5,8,3,2,9,4,1,6,7}
    
70. 70----{5,8,3,2,9,4,7,6,1}
    
71. 71----{5,8,3,4,7,6,1,2,9}
    
72. 72----{5,8,3,4,9,2,1,6,7}
    
73. 73----{5,8,9,2,1,6,7,4,3}
    
74. 74----{5,8,9,2,3,4,1,6,7}
    
75. 75----{5,8,9,2,3,4,7,6,1}
    
76. 76----{5,8,9,4,3,2,1,6,7}
    
77. 77----{5,8,9,4,7,6,1,2,3}
    
78. 78----{7,4,1,6,5,2,3,8,9}
    
79. 79----{7,4,1,6,5,2,9,8,3}
    
80. 80----{7,4,1,6,5,8,3,2,9}
    
81. 81----{7,4,1,6,5,8,9,2,3}
    
82. 82----{7,4,3,2,1,6,5,8,9}
    
83. 83----{7,4,3,2,9,8,5,6,1}
    
84. 84----{7,4,3,8,5,6,1,2,9}
    
85. 85----{7,4,3,8,9,2,1,6,5}
    
86. 86----{7,4,3,8,9,2,5,6,1}
    
87. 87----{7,4,9,2,1,6,5,8,3}
    
88. 88----{7,4,9,2,3,8,5,6,1}
    
89. 89----{7,4,9,8,3,2,1,6,5}
    
90. 90----{7,4,9,8,3,2,5,6,1}
    
91. 91----{7,4,9,8,5,6,1,2,3}
    
92. 92----{7,6,1,2,3,4,9,8,5}
    
93. 93----{7,6,1,2,5,8,3,4,9}
    
94. 94----{7,6,1,2,5,8,9,4,3}
    
95. 95----{7,6,1,2,9,4,3,8,5}
    
96. 96----{7,6,1,4,3,2,5,8,9}
    
97. 97----{7,6,1,4,3,2,9,8,5}
    
98. 98----{7,6,1,4,3,8,5,2,9}
    
99. 99----{7,6,1,4,3,8,9,2,5}
    
100. 100----{7,6,1,4,9,2,3,8,5}
     
101. 101----{7,6,1,4,9,2,5,8,3}
     
102. 102----{7,6,1,4,9,8,3,2,5}
     
103. 103----{7,6,1,4,9,8,5,2,3}
     
104. 104----{7,6,5,2,1,4,3,8,9}
     
105. 105----{7,6,5,2,1,4,9,8,3}
     
106. 106----{7,6,5,2,3,8,9,4,1}
     
107. 107----{7,6,5,2,9,8,3,4,1}
     
108. 108----{7,6,5,8,3,2,1,4,9}
     
109. 109----{7,6,5,8,3,2,9,4,1}
     
110. 110----{7,6,5,8,3,4,1,2,9}
     
111. 111----{7,6,5,8,3,4,9,2,1}
     
112. 112----{7,6,5,8,9,2,1,4,3}
     
113. 113----{7,6,5,8,9,2,3,4,1}
     
114. 114----{7,6,5,8,9,4,1,2,3}
     
115. 115----{7,6,5,8,9,4,3,2,1}
     
116. 116----{9,2,1,4,3,8,5,6,7}
     
117. 117----{9,2,1,4,7,6,5,8,3}
     
118. 118----{9,2,1,6,5,8,3,4,7}
     
119. 119----{9,2,1,6,7,4,3,8,5}
     
120. 120----{9,2,3,8,5,6,1,4,7}
     
121. 121----{9,2,3,8,5,6,7,4,1}
     
122. 122----{9,2,5,8,3,4,1,6,7}
     
123. 123----{9,2,5,8,3,4,7,6,1}
     
124. 124----{9,4,1,2,3,8,5,6,7}
     
125. 125----{9,4,3,8,5,2,1,6,7}
     
126. 126----{9,4,7,6,1,2,3,8,5}
     
127. 127----{9,4,7,6,1,2,5,8,3}
     
128. 128----{9,4,7,6,5,8,3,2,1}
     
129. 129----{9,8,3,2,1,4,7,6,5}
     
130. 130----{9,8,3,2,5,6,1,4,7}
     
131. 131----{9,8,3,2,5,6,7,4,1}
     
132. 132----{9,8,3,4,1,2,5,6,7}
     
133. 133----{9,8,3,4,7,6,1,2,5}
     
134. 134----{9,8,3,4,7,6,5,2,1}
     
135. 135----{9,8,5,2,1,6,7,4,3}
     
136. 136----{9,8,5,2,3,4,1,6,7}
     
137. 137----{9,8,5,2,3,4,7,6,1}
     
138. 138----{9,8,5,6,1,2,3,4,7}
     
139. 139----{9,8,5,6,7,4,1,2,3}
     
140. 140----{9,8,5,6,7,4,3,2,1}

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Treenewbee

1. f[n_] := (p =Select[Permutations@Range@n, AllTrue[Total /@ Partition[#, 2, 1], PrimeQ] &]; {n, Length@p, p}); Array[f, 8, 2]

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天山草

1，2，3，4，5，6，7，8，9, 10  十个数字有 1328 种排列方案，摘录其一部分如下：
1----{1,2,3,4,7,6,5,8,9,10}
2----{1,2,3,4,7,10,9,8,5,6}
3----{1,2,3,4,9,8,5,6,7,10}
4----{1,2,3,4,9,10,7,6,5,8}
5----{1,2,3,8,5,6,7,4,9,10}
...........................................
1326----{10,9,8,5,6,1,2,3,4,7}
1327----{10,9,8,5,6,7,4,1,2,3}
1328----{10,9,8,5,6,7,4,3,2,1}

问题来了：
   ① 对于 \(n≥2\) 个数字，是否总是有解？  
   ② 对于 \(n≥2\) 个数字，方案数 \(a(n)\) 有没有通项计算公式？ 现在已知 \(a(2)=2，a(3)=2，a(4)=8，a(5)=4，a(6)=16，a(7)=24，a(8)=60，a(9)=140，a(10)=1328\)，通项公式是什么？

天山草

在线整数数列大全 OEIS：


我的电脑只能算到 \(a(10)\)，算 \(a(11)\) 死机。按 OEIS，实算 \(a(11)\) 是不是等于 2144？程序代码如下：

1. Clear["Global`*"];
   
2. a = Permutations[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}];
   
3. n = 0;
   
4. Do[c = a[[k]];
   
5. If[PrimeQ [c[[1]] + c[[2]]] && PrimeQ [c[[2]] + c[[3]]] && PrimeQ [c[[3]] + c[[4]]] && PrimeQ [c[[4]] + c[[5]]] && PrimeQ [c[[5]] + c[[6]]] && PrimeQ [c[[6]] + c[[7]]] &&  PrimeQ [c[[7]] + c[[8]]] && PrimeQ [c[[8]] + c[[9]]] &&
   
6.    PrimeQ [c[[9]] + c[[10]]] && PrimeQ [c[[10]] + c[[11]]], n = n + 1;  Print[n, "----", c]], {k, 1, 11!}]

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