Problems & Puzzles: Puzzles

Puzzle 598. A18781

Let's now ask for more terms for an already studied sequence of integers, A187871 by several people.

1, 12, 123, 1234, 14325, 143256, 1234765, 12347658, 123476589, 10123476589

This sequence is defined as the smallest number that is formed by arranging the decimal numbers "1", "2", ..., "n" in some order so that the sum of every pair of adjacent numbers "i" "j" is prime.

Q. Can you send more terms?
 

Contributions came from Fred Schalekamp, Hakan Summakoğlu, Emmanuel Vantieghem, Jan van Delden.

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Fred wrote:

1-10-3-2-11-6-5-8-9-4-7 
10-1-12-11-2-3-4-7-6-5-8-9 
1-10-13-4-3-2-11-12-7-6-5-8-9 
10-1-12-11-2-3-14-5-8-9-4-13-6-7 
1-10-13-4-15-14-3-2-11-12-7-6-5-8-9 
10-1-12-11-2-15-14-3-16-13-4-7-6-5-8-9 
1-10-13-16-15-14-17-12-11-2-3-4-7-6-5-8-9 
10-1-12-11-18-13-16-15-14-17-2-3-4-7-6-5-8-9 
1-10-13-16-15-14-17-12-11-18-19-4-3-2-9-8-5-6-7 
10-1-12-11-18-13-16-15-14-17-20-3-2-9-8-5-6-7-4-19 
1-10-13-16-15-14-17-12-11-18-19-4-3-20-21-2-9-8-5-6-7 
10-1-12-11-18-13-16-15-14-17-20-21-22-19-4-3-2-9-8-5-6-7 
1-10-13-16-15-14-17-12-11-18-19-22-21-20-23-6-5-2-3-8-9-4-7 
10-1-12-11-18-13-16-15-14-17-20-21-22-19-24-23-6-5-2-3-8-9-4-7 
1-10-13-16-15-14-17-12-11-18-19-22-21-20-23-24-5-2-3-8-9-4-25-6-7 
10-1-12-11-18-13-16-15-14-17-20-21-22-19-24-23-8-3-26-5-2-9-4-25-6-7 
1-10-13-16-15-14-17-12-11-18-19-22-21-20-23-24-5-2-27-26-3-8-9-4-25-6-7 
10-1-12-11-18-13-16-15-14-17-20-21-22-19-24-23-6-25-28-3-2-27-26-5-8-9-4-7 
1-10-13-16-15-14-17-12-11-18-19-22-21-20-23-24-29-2-27-26-3-28-25-4-7-6-5-8-9 
10-1-12-11-18-13-16-15-14-17-20-21-2-27-26-3-28-19-22-25-4-7-24-29-30-23-6-5-8-9 
1-10-13-16-15-14-17-12-11-18-19-22-21-20-23-24-29-2-27-26-3-28-25-4-7-30-31-6-5-8-9 
10-1-12-11-18-13-16-15-14-17-20-21-2-27-26-3-28-19-22-25-4-7-24-23-30-29-32-9-8-5-6-31 
1-10-13-16-15-14-17-12-11-18-19-22-21-20-23-24-29-2-27-26-3-28-25-4-33-8-9-32-5-6-31-30-7 
10-1-12-11-18-13-16-15-14-17-20-21-2-27-26-3-28-19-22-25-34-33-4-7-24-23-30-29-32-9-8-5-6-31 
1-10-13-16-15-14-17-12-11-18-19-22-21-20-23-24-29-2-27-26-3-28-25-34-33-4-7-30-31-6-35-32-5-8-9 
10-1-12-11-18-13-16-15-14-17-20-21-2-27-26-3-28-19-22-25-34-33-4-7-24-23-30-29-32-35-36-31-6-5-8-9 
1-10-13-16-15-14-17-12-11-18-19-22-21-20-23-24-29-2-27-26-3-28-25-34-33-4-37-30-31-36-35-32-9-8-5-6-7 
10-1-12-11-18-13-16-15-14-17-20-21-2-27-26-3-28-19-22-25-34-33-38-23-24-29-30-31-36-35-32-5-8-9-4-37-6-7 
1-10-13-16-15-14-17-12-11-18-19-22-21-20-23-24-29-2-27-26-3-28-25-34-33-38-35-32-39-4-37-30-31-36-7-6-5-8-9 
10-1-12-11-18-13-16-15-14-17-20-21-2-27-26-3-28-19-22-25-34-33-38-23-24-29-30-31-36-35-32-39-40-7-4-37-6-5-8-9
...

10-1-12-11-18-13-16-15-14-17-20-21-2-27-26-3-28-19-22-25-34-33-38-23-24-29-30-31-36-35-32-39-40-43-4-37-42-41-48-49-52-45-44-
53-50-47-60-7-46-57-56-51-58-55-54-59-8-63-64-9-62-5-6-61 

***

Hakan wrote:

11) 1 10 3 2 11 6 5 8 9 4 7
12) 10 1 12 11 2 3 4 7 6 5 8 9
13) 1 10 13 4 3 2 11 12 7 6 5 8 9
14) 10 1 12 11 2 3 14 5 8 9 4 13 6 7
15) 1 10 13 4 15 14 3 2 11 12 7 6 5 8 9
16) 10 1 12 11 2 15 14 3 16 13 4 7 6 5 8 9
17) 1 10 13 16 15 14 17 12 11 2 3 4 7 6 5 8 9
18) 10 1 12 11 18 13 16 15 14 17 2 3 4 7 6 5 8 9
19) 1 10 13 16 15 14 17 12 11 18 19 4 3 2 9 8 5 6 7
20) 10 1 12 11 18 13 16 15 14 17 20 3 2 9 8 5 6 7 4 19
21) 1 10 13 16 15 14 17 12 11 18 19 4 3 20 21 2 9 8 5 6 7
22) 10 1 12 11 18 13 16 15 14 17 20 21 22 19 4 3 2 9 8 5 6 7
23) 1 10 13 16 15 14 17 12 11 18 19 22 21 20 23 6 5 2 3 8 9 4 7
24) 10 1 12 11 18 13 16 15 14 17 20 21 22 19 24 23 6 5 2 3 8 9 4 7
25) 1 10 13 16 15 14 17 12 11 18 19 22 21 20 23 24 5 2 3 8 9 4 25 6 7
26) 10 1 12 11 18 13 16 15 14 17 20 21 22 19 24 23 8 3 26 5 2 9 4 25 6 7
27) 1 10 13 16 15 14 17 12 11 18 19 22 21 20 23 24 5 2 27 26 3 8 9 4 25 6 7
28) 10 1 12 11 18 13 16 15 14 17 20 21 22 19 24 23 6 25 28 3 2 27 26 5 8 9 4 7
29) 1 10 13 16 15 14 17 12 11 18 19 22 21 20 23 24 29 2 27 26 3 28 25 4 7 6 5 8 9
30) 10 1 12 11 18 13 16 15 14 17 20 21 22 19 24 23 30 29 2 27 26 3 28 25 4 7 6 5 8 9
31) 1 10 13 16 15 14 17 12 11 18 19 22 21 20 23 24 29 2 27 26 3 28 25 4 7 30 31 6 5 8 9
32) 10 1 12 11 18 13 16 15 14 17 20 21 22 19 24 23 30 29 2 27 26 3 8 5 32 9 28 25 4 7 6 31
33) 1 10 13 16 15 14 17 12 11 18 19 22 21 20 23 24 29 2 27 26 3 28 25 4 33 8 9 32 5 6 31 30 7
34) 10 1 12 11 18 13 16 15 14 17 20 21 22 19 24 23 30 29 2 27 26 3 28 25 34 33 8 5 32 9 4 7 6 31
35) 1 10 13 16 15 14 17 12 11 18 19 22 21 20 23 24 29 2 27 26 3 28 25 34 33 4 7 30 31 6 35 32 5 8 9
36) 10 1 12 11 18 13 16 15 14 17 20 21 22 19 24 23 30 29 2 27 26 3 28 25 34 33 4 7 36 31 6 35 32 5 8 9
37) 1 10 13 16 15 14 17 12 11 18 19 22 21 20 23 24 29 2 27 26 3 28 25 34 33 4 37 30 31 36 35 32 9 8 5 6 7
38) 10 1 12 11 18 13 16 15 14 17 20 21 22 19 24 23 30 29 2 27 26 3 28 25 34 33 38 35 32 5 8 9 4 37 36 31 6 7
39) 1 10 13 16 15 14 17 12 11 18 19 22 21 20 23 24 29 2 27 26 3 28 25 34 33 38 35 32 39 4 37 30 31 36 7 6 5 8 9
40) 10 1 12 11 18 13 16 15 14 17 20 21 22 19 24 23 30 29 2 27 26 3 28 25 34 33 38 35 32 39 40 31 36 37 4 7 6 5 8 9

***

Emmanuel wrote:

1121103476589,
101121123476589,
11013432111276589,
1011211231458941367,
110134151432111276589,
10112112151431613476589,
1101316151417121123476589,
101121118131615141723476589,
11013161514171211181943298567,
1011211181316151417203298567419, 

***

Jan wrote:


11 (1)
1-10-3-2-11-6-5-8-9-4-7

12 (1)
10-1-12-11-2-3-4-7-6-5-8-9

13 (1)
1-10-13-4-3-2-11-12-7-6-5-8-9

14 (1)
10-1-12-11-2-3-14-5-8-9-4-13-6-7

15 (1)
1-10-13-4-15-14-3-2-11-12-7-6-5-8-9

16 (1)
10-1-12-11-2-15-14-3-16-13-4-7-6-5-8-9

17 (1)
1-10-13-16-15-14-17-12-11-2-3-4-7-6-5-8-9

18 (1)
10-1-12-11-18-13-16-15-14-17-2-3-4-7-6-5-8-9

19 (1)
1-10-13-16-15-14-17-12-11-18-19-4-3-2-9-8-5-6-7

20 (1)
10-1-12-11-18-13-16-15-14-17-20-3-2-9-8-5-6-7-4-19

21 (1)
1-10-13-16-15-14-17-12-11-18-19-4-3-20-21-2-9-8-5-6-7

22 (1)
10-1-12-11-18-13-16-15-14-17-20-21-22-19-4-3-2-9-8-5-6-7

23 (1)
1-10-13-16-15-14-17-12-11-18-19-22-21-20-23-6-5-2-3-8-9-4-7

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

25 (1)
1-10-13-16-15-14-17-12-11-18-19-22-21-20-23-24-5-2-3-8-9-4-25-6-7

26 (1)
10-1-12-11-18-13-16-15-14-17-20-21-22-19-24-23-8-3-26-5-2-9-4-25-6-7

27 (1)
1-10-13-16-15-14-17-12-11-18-19-22-21-20-23-24-5-2-27-26-3-8-9-4-25-6-7

28 (1)
10-1-12-11-18-13-16-15-14-17-20-21-22-19-24-23-6-25-28-3-2-27-26-5-8-9-4-7

29 (1)
1-10-13-16-15-14-17-12-11-18-19-22-21-20-23-24-29-2-27-26-3-28-25-4-7-6-5-8-9

30 (1)
10-1-12-11-18-13-16-15-14-17-20-21-22-19-24-23-30-29-2-27-26-3-28-25-4-7-6-5-8
-9

31 (1)
1-10-13-16-15-14-17-12-11-18-19-22-21-20-23-24-29-2-27-26-3-28-25-4-7-30-31-6-5
-8-9

32 (1)
10-1-12-11-18-13-16-15-14-17-20-21-22-19-24-23-30-29-2-27-26-3-8-5-32-9-28-25-4
-7-6-31

33 (1)
1-10-13-16-15-14-17-12-11-18-19-22-21-20-23-24-29-2-27-26-3-28-25-4-33-8-9-32-5
-6-31-30-7

34 (1)
10-1-12-11-18-13-16-15-14-17-20-21-22-19-24-23-30-29-2-27-26-3-28-25-34-33-8-5
-32-9-4-7-6-31

35 (1)
1-10-13-16-15-14-17-12-11-18-19-22-21-20-23-24-29-2-27-26-3-28-25-34-33-4-7-30-31
-6-35-32-5-8-9

36 (1)
10-1-12-11-18-13-16-15-14-17-20-21-22-19-24-23-30-29-2-27-26-3-28-25-34-33-4-7-36
-31-6-35-32-5-8-9

37 (1)
1-10-13-16-15-14-17-12-11-18-19-22-21-20-23-24-29-2-27-26-3-28-25-34-33-4-37-30-31
-36-35-32-9-8-5-6-7

38 (1)
10-1-12-11-18-13-16-15-14-17-20-21-22-19-24-23-30-29-2-27-26-3-28-25-34-33-38-35-32
-5-8-9-4-37-36-31-6-7

39 (1)
1-10-13-16-15-14-17-12-11-18-19-22-21-20-23-24-29-2-27-26-3-28-25-34-33-38-35-32-39
-4-37-30-31-36-7-6-5-8-9

40 (1)
10-1-12-11-18-13-16-15-14-17-20-21-22-19-24-23-30-29-2-27-26-3-28-25-34-33-38-35-32
-39-40-31-36-37-4-7-6-5-8-9

and many more up to 500.

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