"Self-erasure surviving numbers"

[Eric Angelini]

Hello Seqfan and math-fun,

Take an integer like 36, for example.
Concatenate an infinite amount of copies. You get:

  363636363636363636363636363636...

- read the leftmost digit -"3"-,
- jump *over* 3 digits (to the right), land on (3) and erase
  it:

  3636(3)6363636363636363636363636...
  ^
- read the leftmost unread digit, jump only visible digits,
  erase:

  3636(3)6363(6)36363636363636363636...
  ^^
- repeat until you see a substring like this [...(a)36(b)...]
   [(a) and (b) are erased digits - "36" is the integer you
   are, testing]: bingo, you have found a "Self-erasure survi-
   ving number" (SESN):

"36" is such a number:

  3636(3)63(6)3(6)36(3)(6)3(6)36363636363636...
  ^^^^   ^^       ..  <-- hit

This erasing technique gives sometimes quite complicated pat-
terns. "16", for instance, is not a SESN -- but it takes a
while to see:

16(1)616(1)61(6)1(6)(1)6(1)(6)1(6)1(6)1(6)1(6)(1)6(1)616(1)61(6)1(6)
^^   ^^^   ^^   ^      ^      ^   ^   ^   ^      ^   ^^^   ^^   ^
     |_______________________________________________|
                    recurrent pattern

The first SESN I have found by hand are:

0 1 2 3 4 5 6 7 8 9 10 20 23 24 25 26 27 28 29 30 32 36 37
38 39 40 42 ...

[BTW, reading "0" means erasing the closest visible digit
immediately to the right]

No SESN > 10 begins with "1" -- see why?
No SESN > 29 begins with "2", etc.

The sequence is finite, thus.

Last term?

And what about recurrent patterns: do all integers behave
like that? Could some strings be definitely "chaotic"?
I guess no...

Best,
É.