[seqfan] (MAXd+MINd)/2 and iterate
Eric Angelini
Eric.Angelini at kntv.be
Wed Nov 10 17:31:08 CET 2010
Hello SeqFans,
Let's start with n = 723810.
We read n from left to right, digit by digit
in this way:
7 says: replace me by the biggest of the 7 digits
on my right (including me)
Then 7 is replaced by 8 --> n = 823810
2 says: replace me by the biggest of the 2 digits
on my right (including me)
Then 2 is replaced by 3 --> n = 833810
3 says: replace me by the biggest of the 3 digits
on my right (including me)
Then 3 is replaced by 8 --> n = 838810
8 says: replace me by the biggest of the 8 digits
on my right (including me)
Then 8 is replaced by 8 --> n = 838810
1 says: replace me by the biggest of the 1 digit(s)
on my right (including me)
Then 1 is replaced by 1 --> n = 838810
0 says: replace me by the biggest of the 0 digit(s)
on my right (including me)
Then 0 is replaced by nothing and disappears
--> n = 83881
So n = 723810 becomes 83881
This result is called MAXd (maximum digit)
The MINd (minimum digit) operation works the same
way (just replace "biggest" by "smallest", above).
We then get for n: 723810 --> 02101 which is 2101
(no leading zeros allowed).
We will now transform n into n':
n' = (MAXd+MINd)/2
For n = 723810 we have n' = (83881+2101)/2 = 42991
What would happen with n = 1234 and an iteration?
We will present the iteration like this:
/ MAXd / MAXd'
n MAXd+MINd = n' MAXd'+MINd' = n", etc.
\ MINd \ MINd'
/1344 /1899 /1994 /1919 /1919
1234 2578 = 1289 3188 = 1594 3438 = 1719 3038 = 1519 3038 - 1519
\1234 \1289 \1444 \1119 \1119
... we see that 1519 is a fixed point.
What would be S, the "fixed-points" sequence of
(MAXd+MINd)/2 ?
S starts, I think, like this:
S = 1,2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,22, ...
Could we have (MAXd+MINd) = odd number? Yes, in a
(very) few cases:
/3
30 3
\0
If so, the iteration stops immediately as "impossible".
Best,
É.
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