[seqfan] Re: Prime and composite walks concatenated

[Charles Greathouse]

I'm starting to think that neither the original sequence nor its
Cramér analogue is recurrent.  A normal two-dimensional walk comes
close to the origin infinitely often, and since its step sizes are
constant it hits the origin infinitely often.  This sequence probably
(absent Shanks-Rényi type bias) also comes close* to the origin
infinitely often, but because of its increasing step sizes is rather
unlikely to land precisely on the axes.

Would someone more familiar with random walks like to take a stab at
proving this in a random model, either 50% left/50% right or 1/log n
left/otherwise right?

* Relative to its step size.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Mon, Apr 26, 2010 at 10:43 AM, Charles Greathouse
<charles.greathouse at case.edu> wrote:
> The first question is interesting to me: Does the sequence act like a
> random walk?  Is it recurrent?  How does the behavior differ from the
> same sequence over Cramér primes?
>
> The second question is based on the behavior mod 360, which is
> arbitrary and not so interesting to me.
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
> On Mon, Apr 26, 2010 at 10:28 AM, Eric Angelini <Eric.Angelini at kntv.be> wrote:
>>
>> Hello SeqFans,
>> I guess this is old hat -- but cannot find anything on the
>> web or in my books:
>> http://www.cetteadressecomportecinquantesignes.com/PrimeWalk.htm
>>
>> Best,
>> É.
>>
>> ---------
>>
>> (...)
>> Walking instructions:
>>
>> - Start at coordinates (0,0) facing North;
>> - At each stage, you'll have to select the direction and the length of the walk (in steps);
>> - The length of the walk is the same as the preceding one, plus one step;
>> - The direction of the walk depends on its length: if the length is a prime number turn 90° to the left and walk; else turn 90° to the right and walk.
>>
>> A concatenation of the 16 first walks is shown on the sketch above.
>>
>> Question (1):
>> - the 17th walk will pass through the starting point; are there other such walks? Could this be the start (0,1,17,...) of a new sequence for the OEIS?
>>
>> Question (2):
>> - same question, but the direction of the walk is now given by its length "L": L degrees to the left (modulo 360) if L is prime, else L degrees (modulo 360) to the right.
>>
>>
>>
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>>
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>>
>