[seqfan] Re: K(n)—>a(n).

[Allan Wechsler]

Permuting the digits of a number does not change its residue modulo 9. That
explains why 9 always divides the difference between a number and a
permutation of it.

On Tue, Dec 10, 2019 at 9:30 AM David Sycamore via SeqFan <
seqfan at list.seqfan.eu> wrote:

> For integer n let K(n) be the permutation of the digits of n formed by
> sequentially combining the greatest and smallest digits in adjacent pairs
> until running out of digits to play with. Examples: K(1)=1, K(10)=10,
> K(123)= 312, K(277272)=727272, K(539204)=905243.
>
> (Definition is draft, suggestions to improve it are  welcome).
>
> We compute a(n) as follows:
> Take the absolute difference |n-K(n)| to get a new number. Repeat the
> process with that number until reaching a number m for which K(m)=m, then
> a(n)=m.
>
> I get (by hand, so could be some bugs)
>
> 0,1,2,3,4,5,6,7,8,9,10,11,9,9,9,9,9,54,63,72,20,21,22,9,63,9...
>
> Could there be some n for which the above trajectory loops? If so then how
> to define a(n)? (I have not found any such n yet..).
>
> A curious feature seems to be that if a(n) is other than n, then it is a
> multiple of 9, but I have not confirmed the veracity of this.
>
> Sequence does not seem to be in oeis
>
> Any interest in this?
> Best
> David.
>
> ps: Unless I am  mistaken a(123)=63, a(1234)=81, a(4321)=63,
> a(12345)=70434.
>
>
>
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