Start with n = 127. Replace, one by one, every digit ‘d’ of n by ‘d+1’. Iterate.
127 -> 238 -> 349 -> 4510 -> 5621 -> 6732...
*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>?
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
as substring in its orbit under A216556.
> 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,