[seqfan] Re: Strings resurrection

[Maximilian Hasler]

PS: Sorry, it is of course easy to see that

Lemma 1: No number having "20", "30", ..., "90" or "00" as substring
is in the range of A216556,
and therefore A216557(n)=0 for all these numbers.

You may also have noticed that I wanted to write
"the value of A216557(127) (...) could be deduced",
and not "A216557(27)" !

As to that, Eric mentioned that A216556(1016) = 2127
and it seems indeed that

Lemma 2: There is no n such that A216556(n) has 127 as substring, but
n does NOT have 1016 as substring.

Furthermore,

Lemma 3: There is no n such that A216556(n) has 1016 as substring, but
n does NOT have 905 as substring.

Together with Lemma 1, this implies that  A216557(127)=0.
(In similar manner one could reason (and write a program) to detect
all n such that A216557(127)=0.)

Wishing a nice w/e to all SeqFans,

Maximilian


On Sat, Sep 8, 2012 at 12:15 PM, Maximilian Hasler
<maximilian.hasler at gmail.com> wrote:
> Dear Eric, dear SeqFans,
>
> I have created http://oeis.org/A216556 :
> Concatenate decimal digits of n, each increased by 1.
> (approved by Joerg Arndt),
> and http://oeis.org/drafts/A216557 :
> Number of iterations of A216556 until the initial value n appears
> again as a substring; 0 if this will never happen.
> which starts (offset=0):
> 10,9,9,9,9,9,9,9,9,9,9,19,28,37,46,55,64,73,82,90,
>
> Already A216557(20) seems to take an infinite time (>20 sec) to compute,
> maybe s/o can show it's 0 ?
> Should the same be true for n=127 ?
> I found A216557(27)=64 .
> I remarked that that "27" appeas in the 214 digit number
> A216556^{64} (27) =
> 322121102110109211010910998
> 211010910998109989887
> 211010910998109989887
> 1099898879887877687767
> 6657
> 66565547
> 6656554655454437
> 66565546554544365545443544343327
> 665655465545443655454435443433265545443544343325443433243323221.
>
> I think this pattern can be analyzed and the value of A216557(27)
> and other interesting properties about these maps could be deduced.
>
> Maximilian
>
>
> On Sat, Sep 8, 2012 at 10:17 AM, Eric Angelini <Eric.Angelini at kntv.be> wrote:
>>
>> 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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