[seqfan] Copies to the right

[Eric Angelini]

Hello SeqFans,

Let's imagine this procedure for an integer "n" to reproduce itself.

Take "n" = 1234 for instance.

Step 1:
Print 1234 on a squared paper with one digit per cell. By convention we say that the first digit of "n" occupies the cell of rank 1, the second digit of "n" the cell of rank 2, etc.

Example of step 1:
1 2 3 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 

Step 2:
Create an infinite amount of copies of each digit "d" of "n" to the right, with those simple rules:

a) if the digit "d" occupies the cell with rank "c", then the next copies of "d" will successively appear in the cells of rank "c+(d+1)", "c+2(d+1)", "c+3(d+1)", "c+4(d+1)", ... "c+k(d+1)".

b) if a cell is occupied by more than one digit, keep only the biggest one.

Example of step 2 (beginning of the infinite duplication):
1 2 3 4 2 . 3 2 4 . 3 . 1 4 3 . 2 . 4 2 1 . 3 4 1 2 3 . 4 . 3 2 1 4 3 . 1 2 4 . 2 . 3 4 1 . 

Some squares won't be filled, but this is not important. The important question is: when does the initial "n" show again? This "when" will be measured by a "distance" -- which will simply be the amount of cells the first digit of "n" has to "pass through" to reach the cell occupied by its first "sound" copy.

If I'm not wrong, the copy of 1234 is 60 cells away from its original:

1 2 3 4 2 . 3 2 4 . 3 . 1 4 3 . 2 . 4 2 1 . 3 4 1 2 3 . 4 . 3 2 1 4 3 . 1 2 4 . 2 . 3 4 1 . 3 . 4 2 1 . 2 4 3 2 1 . 4 . 1 2 3 4 (hit)

A last remark:
Some integers "vanish" immediately, they cannot reproduce; see for instance what happens to 100:

Step 1:
1 0 0 

Step 2:
1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 ...

"100" has vanished in step 2; there is no "distance" at all here (we could say by convention that its distance is equal to zero)
__________

Ok, the tool is ready. Let's see a few [distances] for "n" to reproduce:

 "n"   [distance]

  0 0  [1]
  1 . 1  [2]
  2 . . 2  [3]
  3 . . . 3  [4]
  4 . . . . 4  [5]
  5 . . . . . 5  [3]
  6 . . . . . . 6  [7]
  7 . . . . . . . 7  [8]
  8 . . . . . . . . 8  [9]
  9 . . . . . . . . . 9  [10]
1 0 1 0  [2]
1 1 1  [1]
1 2 1 . 2 . 1 2  [6]
1 3 1 . 1 3  [4]
1 4 1 . 1 . 4 . 1 . 1 4  [10]
1 5 1 . 1 . 1 5  [6]
1 6 1 . 1 . 1 . 6 . 1 . 1 . 1 6  [14]
1 7 1 . 1 . 1 . 1 7  [8]
1 8 1 . 1 . 1 . 1 . 8 . 1 . 1 . 1 . 1 8  [18]
1 9 1 . 1 . 1 . 1 . 1 9  [10]
2 0 0 2 0  [3]
2 1  [6] (the integers "ab" and "ba" share the same distance)
2 2 . 2 2  [3]
2 3 . 2 . 3 2 . . 3 . . 2 3  [12]
2 4 . 2 . . 4 . . 2 . 4 2 . . 2 4  [15]
2 5 . 2 . . 2 5  [6]
2 6 . 2 . . 2 . 6 2 . . 2 . . 6 . . 2 . . 2 6  [21]
2 7 . 2 . . 2 . . 7 . . 2 . . 2 . 7 2 . . 2 . . 2 7  [24]
2 8 . 2 . . 2 . . 2 8  [9]
2 9 . 2 . . 2 . . 2 . 9 2 . . 2 . . 2 . . 9 . . 2 . . 2 . . 2 9  [30]
3 0 0 0 3 0  [4]
3 1  [4]
3 2  [12]
3 3 . . 3 3  [4]
3 4 . . 3 . 4 . 3 . . 4 3 . . . 4 . . . 3 4  [20]
3 5 . . 3 . . 5 3 . . . 3 5  [12]
3 6 . . 3 . . . 6 . . . 3 . . 6 3 . . . 3 . 6 . 3 . . . 3 6  [28]
3 7 . . 3 . . . 3 7 . . 3 . . . 3 7  [16]
3 8 . . 3 . . . 3 . 8 . 3 . . . 3 . . 8 3 . . . 3 . . . 8 . . . 3 . . . 3 8  [36]
3 9 . . 3 . . . 3 . . 9 3 . . . 3 . . . 3 9  [20]
4 0 0 0 0 4 0  [5]
4 1  [10]
4 2  [15]
4 3  [20]
...
Sequences associated with this idea could be:

- Integers which are their own distance (like 18 or 81)
- Distance(n)
- Smallest integer with distance "n"
- Integers with zero distance...

Best,
É.

http://www.cetteadressecomportecinquantesignes.com/CloneToTheRight.htm