Self-erasing numbers
Eric Angelini
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 > 299 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 not...
Best,
É.
Michael A. Cleverly
> No SESN > 10 begins with "1" -- see why?
Very well could be a simple mistake on my part (which I'll regret posting
about in the morning) but isn't 114 a SESN?
114114114114114114114114114114114114114114114114114114114114
11(4)114114114114114114114114114114114114114114114114114114114
^
11(4)1(1)4114114114114114114114114114114114114114114114114114114
^^
11(4)1(1)4(1)14114114114114114114114114114114114114114114114114114
^^ ^
11(4)1(1)4(1)1411(4)114114114114114114114114114114114114114114114114
^^ ^ ^
11(4)1(1)4(1)14(1)1(4)114114114114114114114114114114114114114114114114
^^ ^ ^ ^
11(4)1(1)4(1)14(1)1(4)114(1)14114114114114114114114114114114114114114114
^^ ^ ^ ^^ !!!
Eric Angelini
^^ ^ ^ ^^ !!!
... Yes, indeed, you're right, Michael, bravo!
I completely overlooked that! I thought that any "1xx" integer
would "self-erase" it's last digit immediately like 1x(x) but
you showed I was wrong! I have to reconsider the ^ fini-
tude of the sequence!
Thanks,
Best
É.