**Strings Resurrection**

Hello SeqFans,

Start with n
= 127. Replace, one by one, every digit ‘d’ of n by ‘d+1’.
Iterate.

127 ->
238 -> 349 -> 4510 -> 5621 -> 6732...

Questions:

*will the
substring <127> reappear at some stage in the iteration of 127?

*If yes,
after how many steps?

*Can we
assign to n=1, n=2, n=3, etc., the number of steps needed to see the substring
< n > reappear in the iteration of n (as defined above)?

*If we go
backwards, we can see that 905 will produce the substring <127> in 2
steps:

905 ->
1016 -> 2127 (hit). Is 905 the smallest integer producing 127?

*What are
the smallest "ancestors" of n=1, n=2, n=3, ...
producing the substring <n>?

Best,

É.

__________

[**Maximilian Hasler**]:

Dear Eric, dear SeqFans,

I have created:

http://oeis.org/A216556 : Concatenate
decimal digits of n, each increased by 1

http://oeis.org/A216557 : Iterations of
A216556 until n reappears as substring

http://oeis.org/A216587 : Preimage of n for A216556

http://oeis.org/A216589 : Numbers n which
don’t have a preimage for A216556

http://oeis.org/A216603 : Indices n for
which A216557(n)=0, i.e., n does
not reappear

as
substring in its orbit under A216556.

[**Giovanni Resta**]:

> will the substring <127> reappear at some
stage in the iteration of 127?

No, it will
not reappear.

In general,
a 0 can only come from a 9, and thus it must always have a 1 in front of it. In
other words it is impossible to have the substring ..x0..
unless x=1.

So ..127.. comes from ..016.. which must be ..1016.. which
comes from ..905.. which is impossible to obtain, since
we have the impossible 90.

__________

Many thanks,
Maximilian and Gianni,

Best,

É.