[seqfan] Fathers and ghosts
Eric Angelini
Eric.Angelini at kntv.be
Sun Dec 29 23:33:27 CET 2013
Hello Seqfans,
Let's pick any positive integer and call it the "father" -- for instance 148.
Lets call 347 the "ghost" produced by 148 according to this first rule:
1 4 8
3 4 7
3 comes from |1-4|
4 comes from |4-8|
7 comes from |8-1|
We produce now a new father f' with the second rule:
--- if f<g then f'=f+g
--- if f>g then f'=f-g
Here we compute f' doing 148+347 as 148<347 --- thus f'=495.
In iterating this procedure, it seems that all integers enter into a loop.
Our starting example, for instance, will produce this sequence of
operations:
1 4 8
+ 3 4 7
----------
4 9 5
+ 5 4 1
----------
1 0 3 6
+ 1 3 3 5
-----------
2 3 7 1
- 1 4 6 1
-----------
9 1 0
- 8 1 9
-------
9 1
- 8 8
-----
3
- 0
---
3 <---- loop
We could compute S, S being the sequence of all integers which are
part of a loop (like 3, seen here);
S starts, I guess, with 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,...
but 16 is not in S as 16 enters into the "5" loop like this:
1 6
+ 5 5
-----
7 1
- 6 6
-----
5
- 0
---
5
The integer 17 is not in S either, as 17 produces this succession of
fathers before entering into a loop:
17--83--28--94--39--105--259--596--165--679--556--545--435--314--83
(loop, see 2nd and last term)
All the above fathers are part of S, of course -- but not 17.
Two remaks:
1) when a ghost starts with a leading zero, we erase this zero
immediately -- example:
1 1 7
- 6 6
--------
5 1
2) is it possible that some integers never end in a loop?
Have a great 2 0 1 4 !
+ 2 1 3 2
-----------
4 1 4 6
- 3 3 2 2
-----------
8 2 4
- 6 2 4
--------
2 0 0
+ 2 0 2
-------
4 0 2
+ 4 2 2
-------
8 2 4 <-- loop!
Best,
É.
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