a(n) > a(n)= 3 ื string(a) ื string(b)
Lets
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)