[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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