[seqfan] Re: Sowing integers

[Maximilian Hasler]

I agree and would it express as follows:
To find the [unique] preimage of [n,s]
where 's' is the position and 'n' an integer of >= s digits,
you successively decrease the digit #s of n and then s
[wrapping from s = 0 to s = #(digits of n)],
until one encounters a digit already equal to zero:
this is the point where the previous sowing started,
and the former value of that zero digit equals the number of steps
(decreased digits) one has just made to come here.

Obviously the sowing cannot have started later (since where the sowing
starts, there is initially left a zero digit), nor earlier (since the
zero digit would then have had to be increased at least by one, if
situated somwhere "after" some earlier starting position, or otherwise
said, it cannot be decreased as to "undo" a hypothetical earlier
sowing step).

I hope I got it correctly...

Maximilian


On Wed, Sep 12, 2012 at 4:01 PM, Max Alekseyev <maxale at gmail.com> wrote:
> Having smaller loop is not possible. If we have certain sequence of
> integers and a position s where next sowing starts, then we can
> uniquely identify the previous sequence (i.e., each sequence/position
> under sowing mapping has not only an unique image but also an unique
> pre-image, and therefore belongs to some loop).
>
> Namely, the position where the previous sowing started is the one with
> the smallest integer in the sequence and (cyclicly) rightmost closest
> to s (possibly s itself) if there are several smallest integers in the
> sequence.
>
> For example, sequence 5,4,3,7,3,5 with the sowing position at the
> integer 4 implies that the previous sowing started at where the
> rightmost integer 3 is currently located and was applied the sequence
> 1,0,0,4,21,1.
>
> Regards,
> Max
>
>
> On Wed, Sep 12, 2012 at 12:49 AM, David Scambler <dscambler at bmm.com> wrote:
>>
>>> Maximilian wrote: " whether (or why) there cannot be some smaller "loop"
>>> in the orbit of some numbers such that the initial position would
>>> never be reproduced"
>>
>> Indeed. I have unsuccessfully searched for short orbits in integers up to 1 million.
>> I am not sure when to stop searching and conjecture instead that there are none.
>>
>> The longest loop so far is 223200 for the integer 98999. Perhaps someone can check this assertion.
>>
>> dave
>>
>>
>>
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