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.

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[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.

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Many thanks, Maximilian and Gianni,

Best,

É.