Copies to the right

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 . (etc.)

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)

Two last remarks:

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).

Some integers don’t vanish – but they don’t copy either, as 1011 and 1012, for instance (thanks to Hans Havermann); their distance is set to zero too:

Step 1:

1 0 1 2

Step 2:

1 0 1 2 1 0 2 0 1 2 1 0 2 0 1 2 1 0 2 0 1 2 1 0 2 0 1 2 1 0 2 ...

__________

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  [8]

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)

- Distance(n)

- Smallest integer with distance “n”

- Integers with zero distance...

Best,

É.

__________

Hans Havermann was quick to answer...

[Hans Havermann]:

> - Distance(n) : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 1, 6, 4, 10, 6, 14, 8, 18, 10, 3, 6, 3, 12, 15, 6, 21, 24, 9, 30, 4, 4, 12, 4, 20, 12, 28, 8, 36, 20, 5, 10, 15, 20, 5, 30, 35, 40, 45, 10, 6, 6, 6, 12, 30, 6, 42, 24, 18, 30, 7, 14, 21, 28, 35, 42, 7, 56, 63, 70, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40, 9, 18, 9, 36, 45, 18, 63, 72, 9, 90, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 0, 2, 6, 4, 10, 6, 14, 8, 18, 10, 0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 0, 6, 6, 12, 30, 6, 42, 24, 18, 30, 0, 4, 12, 4, 20, 12, 28, 8, 36, 20, 0, 10, 30, 20, 10, 30, 70, 40, 90, 10, 0, 6, 6, 12, 30, 6, 42, 24, 18, 30, 0, 14, 42, 28, 70, 42, 14, 56, 126, 70, 0, 8, 24, 8, 40, 24, 56, 8, 72, 40, 0, 18, 18, 36, 90, 18, 126, 72, 18, 90, 0, 10, 30, 20, 10, 30, 70, 40, 90, 10, 3, 6, 3, 12, 15, 6, 21, 24, 9, 30, 6, 3, 6, 12, 30, 6, 42, 24, 18, 30, 3, 6, 1, 12, 15, 6, 21, 24, 9, 30, 12, 12, 12, 12, 60, 12, 84, 24, 36, 60, 15, 30, 15, 60, 15, 30, 105, 120, 45, 30, 6, 6, 6, 12, 30, 6, 42, 24, 18, 30, 21, 42, 21, 84, 105, 42, 21, 168, 63, 210, 24, 24, 24, 24, 120, 24, 168, 24, 72, 120, 9, 18, 9, 36, 45, 18, 63, 72, 9, 90, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 4, 4, 12, 2, 20, 12, 28, 8, 36, 20, 4, 4, 12, 2, 20, 12, 28, 8, 36, 20, 12, 12, 12, 6, 60, 12, 84, 24, 36, 60, 4, 4, 12, 4, 20, 12, 28, 8, 36, 20, 20, 20, 60, 10, 20, 60, 140, 40, 180, 20, 12, 12, 12, 6, 60, 12, 84, 24, 36, 60, 28, 28, 84, 14, 140, 84, 28, 56, 252, 140, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40, 36, 36, 36, 18, 180, 36, 252, 72, 36, 180, 20, 20, 60, 10, 20, 60, 140, 40, 180, 20, 5, 10, 15, 20, 5, 30, 35, 40, 45, 10, 10, 5, 30, 20, 10, 30, 70, 40, 90, 10, 15, 30, 15, 60, 15, 30, 105, 120, 45, 30, 20, 20, 60, 20, 20, 60, 140, 40, 180, 20, 5, 10, 15, 20, 5, 30, 35, 40, 45, 10, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 35, 70, 105, 140, 35, 210, 35, 280, 315, 70, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 45, 90, 45, 180, 45, 90, 315, 360, 45, 90, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 6, 6, 6, 12, 30, 6, 42, 24, 18, 30, 6, 6, 6, 12, 30, 6, 42, 24, 18, 30, 6, 6, 6, 12, 30, 6, 42, 24, 18, 30, 12, 12, 12, 12, 60, 12, 84, 24, 36, 60, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 6, 6, 6, 12, 30, 6, 42, 24, 18, 30, 42, 42, 42, 84, 210, 42, 42, 168, 126, 210, 24, 24, 24, 24, 120, 24, 168, 24, 72, 120, 18, 18, 18, 36, 90, 18, 126, 72, 18, 90, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 7, 14, 21, 28, 35, 42, 7, 56, 63, 70, 14, 7, 42, 28, 70, 42, 14, 56, 126, 70, 21, 42, 21, 84, 105, 42, 21, 168, 63, 210, 28, 28, 84, 28, 140, 84, 28, 56, 252, 140, 35, 70, 105, 140, 35, 210, 35, 280, 315, 70, 42, 42, 42, 84, 210, 42, 42, 168, 126, 210, 7, 14, 21, 28, 35, 42, 7, 56, 63, 70, 56, 56, 168, 56, 280, 168, 56, 56, 504, 280, 63, 126, 63, 252, 315, 126, 63, 504, 63, 630, 70, 70, 210, 140, 70, 210, 70, 280, 630, 70, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40, 24, 24, 24, 24, 120, 24, 168, 24, 72, 120, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 24, 24, 24, 24, 120, 24, 168, 24, 72, 120, 56, 56, 168, 56, 280, 168, 56, 56, 504, 280, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40, 72, 72, 72, 72, 360, 72, 504, 72, 72, 360, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 9, 18, 9, 36, 45, 18, 63, 72, 9, 90, 18, 9, 18, 36, 90, 18, 126, 72, 18, 90, 9, 18, 9, 36, 45, 18, 63, 72, 9, 90, 36, 36, 36, 36, 180, 36, 252, 72, 36, 180, 45, 90, 45, 180, 45, 90, 315, 360, 45, 90, 18, 18, 18, 36, 90, 18, 126, 72, 18, 90, 63, 126, 63, 252, 315, 126, 63, 504, 63, 630, 72, 72, 72, 72, 360, 72, 504, 72, 72, 360, 9, 18, 9, 36, 45, 18, 63, 72, 9, 90, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 20, 20, 60, 20, 20, 60, 140, 40, 180, 20, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 70, 70, 210, 140, 70, 210, 70, 280, 630, 70, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 4, 10, 6, 14, 8, 18, 10, 6, 0, 0, 12, 30, 6, 42, 24, 18, 30, 4, 0, 0, 4, 20, 12, 28, 8, 36, 20, 10, 0, 0, 20, 10, 30, 70, 40, 90, 10, 6, 0, 0, 12, 30, 6, 42, 24, 18, 30, 14, 0, 0, 28, 70, 42, 14, 56, 126, 70, 8, 0, 0, 8, 40, 24, 56, 8, 72, 40, 18, 0, 0, 36, 90, 18, 126, 72, 18, 90, 10, 0, 0, 20, 10, 30, 70, 40, 90, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 4, 5, 6, 7, 8, 9, 10, 0, 3, 0, 12, 15, 6, 21, 24, 9, 30, 0, 4, 0, 4, 20, 12, 28, 8, 36, 20, 0, 5, 0, 20, 5, 30, 35, 40, 45, 10, 0, 6, 0, 12, 30, 6, 42, 24, 18, 30, 0, 7, 0, 28, 35, 42, 7, 56, 63, 70, 0, 8, 0, 8, 40, 24, 56, 8, 72, 40, 0, 9, 0, 36, 45, 18, 63, 72, 9, 90, 0, 10, 0, 20, 10, 30, 70, 40, 90, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 3, 0, 12, 30, 6, 42, 24, 18, 30, 6, 3, 0, 12, 30, 6, 42, 24, 18, 30, 12, 12, 0, 12, 60, 12, 84, 24, 36, 60, 30, 15, 0, 60, 30, 30, 210, 120, 90, 30, 6, 6, 0, 12, 30, 6, 42, 24, 18, 30, 42, 21, 0, 84, 210, 42, 42, 168, 126, 210, 24, 24, 0, 24, 120, 24, 168, 24, 72, 120, 18, 9, 0, 36, 90, 18, 126, 72, 18, 90, 30, 30, 0, 60, 30, 30, 210, 120, 90, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 2, 20, 12, 28, 8, 36, 20, 12, 12, 0, 6, 60, 12, 84, 24, 36, 60, 4, 4, 0, 4, 20, 12, 28, 8, 36, 20, 20, 20, 0, 10, 20, 60, 140, 40, 180, 20, 12, 12, 0, 6, 60, 12, 84, 24, 36, 60, 28, 28, 0, 14, 140, 84, 28, 56, 252, 140, 8, 8, 0, 8, 40, 24, 56, 8, 72, 40, 36, 36, 0, 18, 180, 36, 252, 72, 36, 180, 20, 20, 0, 10, 20, 60, 140, 40, 180, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 5, 0, 20, 10, 30, 70, 40, 90, 10, 30, 15, 0, 60, 30, 30, 210, 120, 90, 30, 20, 20, 0, 20, 20, 60, 140, 40, 180, 20, 10, 5, 0, 20, 10, 30, 70, 40, 90, 10, 30, 30, 0, 60, 30, 30, 210, 120, 90, 30, 70, 35, 0, 140, 70, 210, 70, 280, 630, 70, 40, 40, 0, 40, 40, 120, 280, 40, 360, 40, 90, 45, 0, 180, 90, 90, 630, 360, 90, 90, 10, 10, 0, 20, 10, 30, 70, 40, 90, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 6, 0, 12, 30, 6, 42, 24, 18, 30, 6, 6, 0, 12, 30, 6, 42, 24, 18, 30, 12, 12, 0, 12, 60, 12, 84, 24, 36, 60, 30, 30, 0, 60, 30, 30, 210, 120, 90, 30, 6, 6, 0, 12, 30, 6, 42, 24, 18, 30, 42, 42, 0, 84, 210, 42, 42, 168, 126, 210, 24, 24, 0, 24, 120, 24, 168, 24, 72, 120, 18, 18, 0, 36, 90, 18, 126, 72, 18, 90, 30, 30, 0, 60, 30, 30, 210, 120, 90, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 7, 0, 28, 70, 42, 14, 56, 126, 70, 42, 21, 0, 84, 210, 42, 42, 168, 126, 210, 28, 28, 0, 28, 140, 84, 28, 56, 252, 140, 70, 35, 0, 140, 70, 210, 70, 280, 630, 70, 42, 42, 0, 84, 210, 42, 42, 168, 126, 210, 14, 7, 0, 28, 70, 42, 14, 56, 126, 70, 56, 56, 0, 56, 280, 168, 56, 56, 504, 280, 126, 63, 0, 252, 630, 126, 126, 504, 126, 630, 70, 70, 0, 140, 70, 210, 70, 280, 630, 70, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 8, 0, 8, 40, 24, 56, 8, 72, 40, 24, 24, 0, 24, 120, 24, 168, 24, 72, 120, 8, 8, 0, 8, 40, 24, 56, 8, 72, 40, 40, 40, 0, 40, 40, 120, 280, 40, 360, 40, 24, 24, 0, 24, 120, 24, 168, 24, 72, 120, 56, 56, 0, 56, 280, 168, 56, 56, 504, 280, 8, 8, 0, 8, 40, 24, 56, 8, 72, 40, 72, 72, 0, 72, 360, 72, 504, 72, 72, 360, 40, 40, 0, 40, 40, 120, 280, 40, 360, 40, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 9, 0, 36, 90, 18, 126, 72, 18, 90, 18, 9, 0, 36, 90, 18, 126, 72, 18, 90, 36, 36, 0, 36, 180, 36, 252, 72, 36, 180, 90, 45, 0, 180, 90, 90, 630, 360, 90, 90, 18, 18, 0, 36, 90, 18, 126, 72, 18, 90, 126, 63, 0, 252, 630, 126, 126, 504, 126, 630, 72, 72, 0, 72, 360, 72, 504, 72, 72, 360, 18, 9, 0, 36, 90, 18, 126, 72, 18, 90, 90, 90, 0, 180, 90, 90, 630, 360, 90, 90, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 10, 0, 20, 10, 30, 70, 40, 90, 10, 30, 30, 0, 60, 30, 30, 210, 120, 90, 30, 20, 20, 0, 20, 20, 60, 140, 40, 180, 20, 10, 10, 0, 20, 10, 30, 70, 40, 90, 10, 30, 30, 0, 60, 30, 30, 210, 120, 90, 30, 70, 70, 0, 140, 70, 210, 70, 280, 630, 70, 40, 40, 0, 40, 40, 120, 280, 40, 360, 40, 90, 90, 0, 180, 90, 90, 630, 360, 90, 90, 10, 10, 0, 20, 10, 30, 70, 40, 90, 10, 0, 0, 3, 12, 15, 6, 21, 24, 9, 30, 0, 0, 6, 12, 30, 6, 42, 24, 18, 30, 0, 0, 3, 12, 15, 6, 21, 24, 9, 30, 0, 0, 12, 12, 60, 12, 84, 24, 36, 60, 0, 0, 15, 60, 15, 30, 105, 120, 45, 30, 0, 0, 6, 12, 30, 6, 42, 24, 18, 30, 0, 0, 21, 84, 105, 42, 21, 168, 63, 210, 0, 0, 24, 24, 120, 24, 168, 24, 72, 120, 0, 0, 9, 36, 45, 18, 63, 72, 9, 90, 0, 0, 30, 60, 30, 30, 210, 120, 90, 30, 0, 0, 6, 12, 30, 6, 42, 24, 18, 30, 0, 0, 3, 12, 15, 6, 21, 24, 9, 30, 0, 0, 6, 12, 30, 6, 42, 24, 18, 30, 0, 0, 12, 12, 60, 12, 84, 24, 36, 60, 0, 0, 30, 60, 30, 30, 210, 120, 90, 30, 0, 0, 6, 12, 30, 6, 42, 24, 18, 30, 0, 0, 42, 84, 210, 42, 42, 168, 126, 210, 0, 0, 24, 24, 120, 24, 168, 24, 72, 120, 0, 0, 18, 36, 90, 18, 126, 72, 18, 90, 0, 0, 30, 60, 30, 30, 210, 120, 90, 30, 0, 0, 3, 12, 15, 6, 21, 24, 9, 30, 0, 0, 6, 12, 30, 6, 42, 24, 18, 30, 0, 0, 1, 4, 5, 6, 7, 8, 9, 10, 0, 0, 12, 12, 60, 12, 84, 24, 36, 60, 0, 0, 15, 60, 15, 30, 105, 120, 45, 30, 0, 0, 6, 12, 30, 6, 42, 24, 18, 30, 0, 0, 21, 84, 105, 42, 21, 168, 63, 210, 0, 0, 24, 24, 120, 24, 168, 24, 72, 120, 0, 0, 9, 36, 45, 18, 63, 72, 9, 90, 0, 0, 30, 60, 30, 30, 210, 120, 90, 30, 0, 0, 12, 6, 60, 12, 84, 24, 36, 60, 0, 0, 12, 6, 60, 12, 84, 24, 36, 60, 0, 0, 12, 6, 60, 12, 84, 24, 36, 60, 0, 0, 12, 12, 60, 12, 84, 24, 36, 60, 0, 0, 60, 30, 60, 60, 420, 120, 180, 60, 0, 0, 12, 6, 60, 12, 84, 24, 36, 60, 0, 0, 84, 42, 420, 84, 84, 168, 252, 420, 0, 0, 24, 24, 120, 24, 168, 24, 72, 120, 0, 0, 36, 18, 180, 36, 252, 72, 36, 180, 0, 0, 60, 30, 60, 60, 420, 120, 180, 60, 0, 0, 15, 60, 15, 30, 105, 120, 45, 30, 0, 0, 30, 60, 30, 30, 210, 120, 90, 30, 0, 0, 15, 60, 15, 30, 105, 120, 45, 30, 0, 0, 60, 60, 60, 60, 420, 120, 180, 60, 0, 0, 15, 60, 15, 30, 105, 120, 45, 30, 0, 0, 30, 60, 30, 30, 210, 120, 90, 30, 0, 0, 105, 420, 105, 210, 105, 840, 315, 210, 0, 0, 120, 120, 120, 120, 840, 120, 360, 120, 0, 0, 45, 180, 45, 90, 315, 360, 45, 90, 0, 0, 30, 60, 30, 30, 210, 120, 90, 30, 0, 0, 6, 12, 30, 6, 42, 24, 18, 30, 0, 0, 6, 12, 30, 6, 42, 24, 18, 30, 0, 0, 6, 12, 30, 6, 42, 24, 18, 30, 0, 0, 12, 12, 60, 12, 84, 24, 36, 60, 0, 0, 30, 60, 30, 30, 210, 120, 90, 30, 0, 0, 6, 12, 30, 6, 42, 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40, 40, 120, 280, 40, 360, 40, 120, 120, 120, 120, 120, 120, 840, 120, 360, 120, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 120, 120, 120, 120, 120, 120, 840, 120, 360, 120, 280, 280, 840, 280, 280, 840, 280, 280, 2520, 280, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 360, 360, 360, 360, 360, 360, 2520, 360, 360, 360, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 180, 180, 180, 180, 180, 180, 1260, 360, 180, 180, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 630, 630, 630, 1260, 630, 630, 630, 2520, 630, 630, 360, 360, 360, 360, 360, 360, 2520, 360, 360, 360, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 20, 20, 60, 20, 20, 60, 140, 40, 180, 20, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 30, 30, 30, 60, 30, 30, 210, 120, 90, 30, 70, 70, 210, 140, 70, 210, 70, 280, 630, 70, 40, 40, 120, 40, 40, 120, 280, 40, 360, 40, 90, 90, 90, 180, 90, 90, 630, 360, 90, 90, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, ...

> Integers with zero distance : 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 1000, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1009, 1011, 1012, 1021, 1022, 1031, 1032, 1041, 1042, 1051, 1052, 1061, 1062, 1071, 1072, 1081, 1082, 1091, 1092, 1100, 1101, 1102, 1103, 1104, 1105, 1106, 1107, 1108, 1109, 1110, 1112, 1120, 1122, 1130, 1132, 1140, 1142, 1150, 1152, 1160, 1162, 1170, 1172, 1180, 1182, 1190, 1192, 1200, 1201, 1202, 1203, 1204, 1205, 1206, 1207, 1208, 1209, 1212, 1222, 1232, 1242, 1252, 1262, 1272, 1282, 1292, 1300, 1301, 1302, 1303, 1304, 1305, 1306, 1307, 1308, 1309, 1312, 1322, 1332, 1342, 1352, 1362, 1372, 1382, 1392, 1400, 1401, 1402, 1403, 1404, 1405, 1406, 1407, 1408, 1409, 1412, 1422, 1432, 1442, 1452, 1462, 1472, 1482, 1492, 1500, 1501, 1502, 1503, 1504, 1505, 1506, 1507, 1508, 1509, 1512, 1522, 1532, 1542, 1552, 1562, 1572, 1582, 1592, 1600, 1601, 1602, 1603, 1604, 1605, 1606, 1607, 1608, 1609, 1612, 1622, 1632, 1642, 1652, 1662, 1672, 1682, 1692, 1700, 1701, 1702, 1703, 1704, 1705, 1706, 1707, 1708, 1709, 1712, 1722, 1732, 1742, 1752, 1762, 1772, 1782, 1792, 1800, 1801, 1802, 1803, 1804, 1805, 1806, 1807, 1808, 1809, 1812, 1822, 1832, 1842, 1852, 1862, 1872, 1882, 1892, 1900, 1901, 1902, 1903, 1904, 1905, 1906, 1907, 1908, 1909, 1912, 1922, 1932, 1942, 1952, 1962, 1972, 1982, 1992, 2000, 2001, 2010, 2011, 2020, 2021, 2030, 2031, 2040, 2041, 2050, 2051, 2060, 2061, 2070, 2071, 2080, 2081, 2090, 2091, 2100, 2101, 2110, 2111, 2120, 2121, 2130, 2131, 2140, 2141, 2150, 2151, 2160, 2161, 2170, 2171, 2180, 2181, 2190, 2191, 2200, 2201, 2210, 2211, 2220, 2221, 2230, 2231, 2240, 2241, 2250, 2251, 2260, 2261, 2270, 2271, 2280, 2281, 2290, 2291, 2300, 2301, 2310, 2311, 2320, 2321, 2330, 2331, 2340, 2341, 2350, 2351, 2360, 2361, 2370, 2371, 2380, 2381, 2390, 2391, 2400, 2401, 2410, 2411, 2420, 2421, 2430, 2431, 2440, 2441, 2450, 2451, 2460, 2461, 2470, 2471, 2480, 2481, 2490, 2491, 2500, 2501, 2510, 2511, 2520, 2521, 2530, 2531, 2540, 2541, 2550, 2551, 2560, 2561, 2570, 2571, 2580, 2581, 2590, 2591, 2600, 2601, 2610, 2611, 2620, 2621, 2630, 2631, 2640, 2641, 2650, 2651, 2660, 2661, 2670, 2671, 2680, 2681, 2690, 2691, 2700, 2701, 2710, 2711, 2720, 2721, 2730, 2731, 2740, 2741, 2750, 2751, 2760, 2761, 2770, 2771, 2780, 2781, 2790, 2791, 2800, 2801, 2810, 2811, 2820, 2821, 2830, 2831, 2840, 2841, 2850, 2851, 2860, 2861, 2870, 2871, 2880, 2881, 2890, 2891, 2900, 2901, 2910, 2911, 2920, 2921, 2930, 2931, 2940, 2941, 2950, 2951, 2960, 2961, 2970, 2971, 2980, 2981, 2990, 2991, 3001, 3011, 3021, 3031, 3041, 3051, 3061, 3071, 3081, 3091, 3100, 3110, 3120, 3130, 3140, 3150, 3160, 3170, 3180, 3190, 4001, 4011, 4021, 4031, 4041, 4051, 4061, 4071, 4081, 4091, 4100, 4110, 4120, 4130, 4140, 4150, 4160, 4170, 4180, 4190, 5001, 5011, 5021, 5031, 5041, 5051, 5061, 5071, 5081, 5091, 5100, 5110, 5120, 5130, 5140, 5150, 5160, 5170, 5180, 5190, 6001, 6011, 6021, 6031, 6041, 6051, 6061, 6071, 6081, 6091, 6100, 6110, 6120, 6130, 6140, 6150, 6160, 6170, 6180, 6190, 7001, 7011, 7021, 7031, 7041, 7051, 7061, 7071, 7081, 7091, 7100, 7110, 7120, 7130, 7140, 7150, 7160, 7170, 7180, 7190, 8001, 8011, 8021, 8031, 8041, 8051, 8061, 8071, 8081, 8091, 8100, 8110, 8120, 8130, 8140, 8150, 8160, 8170, 8180, 8190, 9001, 9011, 9021, 9031, 9041, 9051, 9061, 9071, 9081, 9091, 9100, 9110, 9120, 9130, 9140, 9150, 9160, 9170, 9180, 9190, ...

> Integers which are their own distance (like 18)

> Smallest integer with distance "n"

These might end up being difficult/impossible to extend. I only have 49 distinct distances < 10000 in my range:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 56, 60, 63, 70, 72, 84, 90, 105, 120, 126, 140, 168, 180, 210, 252, 280, 315, 360, 420, 504, 630, 840, 1260, 2520.

A points-joined graph:

Many thanks, Hans, for the corrections, results and nice graph!

__________

October 11th 2012, update:

Jean-Paul Davalan, answering me (Eric) on fr.sci.maths

[Eric]:

> Bonjours à tous,

il semblerait que 18 soit le seul entier inférieur à 10.000 qui soit sa propre distance (de clonage) au sens expliqué ici. Confirmez-vous ? Lui trouverez-vous des compagnons au delà de 10.000 ?

[Jean-Paul]:

> Bonjour,

cette distance, c’est pas tout simplement le ppcm des d+1 (les d sont les chiffres décimaux) ?

Par exemple, pour 18, le ppcm de 2=1+1 et de 9=8+1 est 18 ; le ppcm de tous les d+1 que sont 2, 3, 4, ..., 9, 10 est 8*9*5*7 = 2520, c’est le maximum possible, et l’ensemble des diviseurs de 2520 est 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 56, 60, 63, 70, 72, 84, 90, 105, 120, 126, 140, 168, 180, 210, 252, 280, 315, 360, 420, 504, 630, 840, 1260, 2520 ce qui donne, en y ajoutant la valeur 0, la liste des 49 distances que tu as sur ta page, Éric. Il n’y en a pas d’autres, si ?

[Eric, to Hans Havermann, 15 days later]:

> What do you think, Hans? Are there other distances possible? I guess not, Jean-Paul’s remark seems sound...

[Hans Havermann]:

> Yes, it’s a clever observation. I’ll do a 24-hour run past 10000 to convince myself that there are no counterexamples.

(...)

In the range 10000-99999, I looked for distances < 12000 other than the 49 we knew. Nothing.

__________

So, Jean-Paul has the answer: there are no other integers than 18 which are their own distance.

Thanks to all contributors,

Best,

É.