a(n) > a(n)= 3 string(a) string(b)

Lets see what the two rules hereunder (and the split technique) produce:

a) if a(n) has one single digit, then a(n)>a(n)= [3ืa(n)]; example: 7 >[3ื7] = 21;

b) if a(n) has more than one digit then we split a(n) into two strings s1 and s2 such that a(n)>a(n)= [3ืs1ืs2]

How to split:

- for a two-digit number [t,u] we have s1=t and s2=u (example: 28 splits into 2 and 8)

- for a three, four, five...-digit number we take all possible ways of partitioning a(n) into two strings (not allowing string s2 to have a leading zero). Example for a(n)=123405:

123405 --> s1=1 and s2=23405

or --> s1=12 and s2=3405

or --> s1=123 and s2=405

or --> s1=12340 and s2=5

We do not accept the split s1=1234 and s2=05

So, if a(n)= 123405 we have a(n)= 3ื1ื23405 =  70215

or a(n)= 3ื12ื3405 = 122580

or a(n)= 3ื123ื405 = 149445

or a(n)= 3ื12340ื5 = 185100

We then make a(n)=a(n) and iterate the process.

__________

Here is a quick table for a(n)= 0, 1, 2, 3, 4, ... 99, 100:

0 > (3ื0)= 0 = fixed point

1 > 3 > 9 > 27 > (3ื2ื7)= 42 > (3ื4ื2)= 24 > (3ื2ื4)= 24 = fixed point

2 > 6 > 18 > (3ื1ื8)= 24

3 ...

4 > 12 > 6 ...

5 > 15 > (3ื1ื5)= 15 = fixed point

6 ...

7 > 21 > 6 ...

8 > 24

9 > ...

10 > 0 ...

11 > 3 ...

12 > ...

13 > 9 ...

14 > 12 ...

15

16 > 18 ...

17 > 21 > 6 ...

18 > ...

19 > 27 ...

20 > 0

21 > (see 12)

22 > 12 ...

23 > 18 ...

24

...

27 > ...

28 > 48 > 96 > 162 >(3ื1ื62)= 186 >(3ื1ื86)= 258 >(3ื2ื58)= 348 >(3ื3ื48)= 432 >(3ื4ื32)= 384 > etc.

432 >(3ื43ื2)= 258 (loop)

348 >(3ื34ื8)= 816 >(3ื8ื16)= 384 ...

816 >(3ื81ื6)= 1458 > etc.

258 >(3ื25ื8)= 600 >(3ื60ื0)= 0

186 >(3ื18ื6)= 324 >(3ื3ื24)= 216 >(3ื2ื16)= 96 (loop)

216 >(3ื21ื6)= 378 >(3ื3ื78)= 702 > etc.

378 >(3ื37ื8)= 888 > etc.

162 >(3ื16ื2)= 96 (loop)

29 > 54 > 60 > 0

30 > 0

31 > (see 13)

32 > (see 23)

33 > 27 ...

34 > 36 > 54 ...

35 > 45 > 60 ...

36 > 54 ...

37 > 63 > 54 ...

38 > 72 > 42 > 24 ...

39 > 81 > 24 ...

40 > 0

41 > (see 14)

42 > (see 24)

43 > (see 34)

44 > 48 ...

45 > 60 ...

46 > 72 > 42 ...

47 > 84 > 96 ...

48 > 96 ...

49 > 108 >(3ื10ื8)= 240 >(3ื2ื40)= 240 = fixed point

50 > 0

51 (see 15)

...

55 > 75 > 105 >(3ื10ื5)= 150 >(3ื1ื50)= 150 = fixed point

150 >(3ื15ื0)= 0

56 > 90 > 0

57 > 105 ...

58 > 120 > 60 ...

59 > 135 >(3ื1ื35)= 105 ...

135 >(3ื13ื5)= 195 >(3ื1ื95)= 285 >(3ื2ื85)= 510 >(3ื5ื10)= 150

510 >(3ื51ื0)= 0

285 >(3ื28ื5)= 420 >(3ื4ื20)= 240

195 >(3ื19ื5)= 285 ...

60 ...

...

66 > 108 ...

67 > 126 >(3ื1ื26)= 78 > 168 >(3ื1ื68)= 204 >(3ื20ื4)= 240

168 >(3ื16ื8)= 384 ...

126 >(3ื12ื6)= 216 ...

68 > 144 >(3ื1ื44)= 132 >(3ื1ื32)= 96 ...

132 >(3ื13ื2)= 78 ...

144 >(3ื14ื4)= 168 ...

69 > 162 ...

70 > 0

...

77 > 147 >(3ื1ื47)= 141 >(3ื1ื41)= 123 >(3ื1ื23)= 69 ...

123 >(3ื12ื3)= 108 ...

141 >(3ื14ื1)= 42 ...

147 >(3ื14ื7)= 294 >(3ื2ื94)= 564 >(3ื5ื64)= 960 >(3ื9ื60)= 1620 >(3ื1ื620)= 1860 >(3ื1ื860) = 2580 > etc.

1860 >(3ื18ื60) = 3240 > etc.

1620 >(3ื16ื20)= 960 >(3ื9ื60)= 1080 > etc.

960 >(3ื96ื0)= 0

564 >(3ื56ื4)= 672 >(3ื6ื72)= 1296 >(3ื1ื296)= 888 ...

1296 >(3ื12ื96)= 3456 > etc.

1296 >(3ื129ื6)= 2322 > etc.

672 >(3ื67ื2)= 402 >(3ื40ื2)= 240

294 >(3ื29ื4)= 348 ...

78 > 168 ...

79 > 189 >(3ื1ื89)= 267 >(3ื2ื67)= 402 ...

267 >(3ื26ื7)= 546 >(3ื5ื46)= 690 >(3ื6ื90)= 1620 ...

189 >(3ื18ื9)= 486 >(3ื4ื86)= 1032 >(3ื10ื32)= 960 ...

1032 >(3ื103ื2)= 618 >(3ื6ื18)= 324 ...

618 >(3ื61ื8)= 1464 >(3ื1ื464)= 1392 > etc.

1464 >(3ื14ื64)= 2688 > etc.

486 >(3ื48ื6)= 864 >(3ื8ื64)= 1536 >(3ื1ื536)= 1608 >(3ื1ื608)= 1824 > etc.

1608 >(3ื160ื8)= 3840 > etc.

80 > 0

...

88 > 192 >(3ื1ื92)= 276 >(3ื2ื76)= 456 ...

276 >(3ื27ื6)= 486 ...

192 >(3ื19ื2)= 114 >(3ื1ื14)= 42 ...

114 >(3ื11ื4)= 132 ...

89 > 216 ...

90 > 0

...

99 > 243 >(3ื2ื43)= 258 ...

243 >(3ื24ื3)= 216 ...

100 > 0

etc.

Questions:

1) What would be the list of the fixed points? Seems that it should start with 0, 15, 24,... What about 150 and 240? Those integers are fixed points... following one path of their tree... (the other leads to zero)

2) What would be the list of the smallest numbers part of a loop (again, if we choose accordingly a path in the tree)? Does this list start with, 0, 15, 24, 69, 150, ...

3) Is 28 the first integer producing an infinite sequence? What would be the lexicographically first infinite path? Does this have a sense?

Best,

ษ. > (3ืษืric) or (3ืษrืic) or (3ืษriืc)