[seqfan] Re: Riecaman

Thu Aug 29 14:24:13 CEST 2019

Hello Hugo, Neil and others,

a few observations and results:

1. Perhaps the unexpected appearance of the 9 as the quotient limit 
becomes a bit less mysterious when you consider ALL "local minima" (as 
opposed to only those for which the trajectory value is even), in which 
case the quotients seem to approach a limit of 3 - that looks a lot less 
surprising!

2. Of course you can compute a trajectory even for starting values which 
at some point reach zero - you just keep going the same way (two adds, 
one subtract etc.) ad infinitum. It comes as no surprise that the 
quotients of the local minima seem to approach 3 here as well.

3. To generalize the construction even further, we can distinguish 
between adding and subtracting by forcing the trajectory to never fall 
below a fixed threshold (the original definition will then have a 
threshold of 0). Again no surprises here, the local minima quotients 
still converge to 3.

4. What surprises me a lot more is that the same behavior can be 
observed in all the sequences suggested by Neil (with the exception of 
round(sqrt(n)), i.e. A000194, for which the trajectories look 
different): In all these cases the quotients seem to approach 3. So, 
there seems to be something at work here that does not necessarily have 
anything to do with primes!

Regards
Jens

Am 2019-08-29 04:29, schrieb Neil Sloane:
> Hugo,  That's a lovely discovery!  I would really like to see the 
> graphs of
> some of these trajectories, to get an idea of how irregular (or 
> fractal)
> they are.
> Could you create sequences for the "primitive" trajectories 6 20 50 51 
> 70
> 71 say?
> With good fat b-files?
> 
> (I could do it, but they are your sequences.)
> 
> The first trajectory, of 6, starts
> 
> 6, 4, 1, 6, 13, 2, 15, 32, 13, 36, 7, 38, 1, 42, 85, 38, 91, 32, 93, 
> 26,
> 97, 24, 103, 20, 109, 12, 113, 10, 117, 8, 121, 248, 117, 254, 115, 
> 264,
> 113, 270, 107, 274, 101, 280, 99, 290, 97, 294, 95, 306, 83, 310, 81, 
> 314,
> 75, 316, 65, 322, 59, 328, 57, 334, 53, 336, 43, 350, 39, 352, 35, 366, 
> 29,
> 376, 27, 380, 21, 388, 15, 394, 11, 400, 3, 404, 813, ...
> 
> which is a bit like Recaman (except there we forbid getting a repeat 
> when
> we subtract, so there is a lot more adding than you have, with 
> resulting
> high peaks and rare big drops).
> 
> So where does your factor of 9 come from?  One way to investigate that
> would be to try
> your procedure, but basing it on some other sequence than the primes.
> 
> Presumably the lucky numbers - which grow like the primes - would also 
> give
> a factor of 9.
> What about numbers that are the product of exactly 2 primes, A001358, 
> which
> grow a bit more slowly than the primes? What about the triangular 
> numbers?
> What about using round(sqrt(n)) as the controlling sequence? And so on.
> 
> Nice problem!
> 
> Best regards
> Neil
> 
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, 
> NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
> 
> 
> 
> On Wed, Aug 28, 2019 at 8:46 PM <hv at crypt.org> wrote:
> 
>> Here is a Recaman-inspired sequence with apparent ties to Riemann.
>> 
>> The main purpose of this is to ask why on earth we end up with a 
>> series
>> of primes whose successive ratios converge to 9.
>> 
>> Define a mapping n_{i-1} -> n_i as:
>>   n_{i-1} + p_i    if p_i > n_{i-1}
>>   n_{i-1} - p_i    otherwise
>> with a given starting point n_0, and where p_i is the i'th prime.
>> 
>> Define a(n) as the least positive k such that n_k is 0 when we set n_0 
>> = n,
>> or as 0 if no such k exists.
>> 
>> I've calculated most values of a(0) .. a(100), (see below); the 
>> missing
>> ones
>> are for n in { 6 16 20 30 42 50 51 56 70 71 76 84 85 90 92 }, and 
>> there
>> things get interesting. If k exists for any of these, it is at least
>> 1.5e10.
>> 
>> Several of those hard ones collapse to identical trajectories early 
>> on:
>>   6, 16, 30, 56, 90
>>   20, 42, 76
>>   50, 84
>>   51, 85
>> .. so a(6) = a(16) etc. Taking the first of such sets as "primitive",
>> that leaves primitives { 6 20 50 51 70 71 92 }.
>> 
>> The shape of the trajectories is that we alternately add and subtract,
>> with the net effect that n_{i+2} is smaller than n_i by the prime
>> difference,
>> until we reach a local mininum that is either zero (terminating the
>> process) or too small for the prime difference causing us to add twice
>> in a row. When we hit a local minimum we also switch the parity of
>> the lower of each pair of terms, and obviously we can't hit zero when
>> that's odd.
>> 
>> Checking the even local minima for the trajectory of 6, for example,
>> gives:
>> n  n_i  i          p_i
>> 6    2  5          13
>> 6    8  29         113
>> 6    4  199        1223
>> 6    2  1355       11197
>> 6    8  9589       99971
>> 6   30  70579      890377
>> 6   60  539961     8002847
>> 6    4  4228745    72001673
>> 6   38  33690443   647909833
>> 6   16  272003821  5830319399
>> 6   16  2219823175 52470123707
>> 
>> I looked at this mostly to try and understand whether I should expect
>> 0 values of the sequence to exist - my conjecture is no - but looking
>> at the ratio of the p_i for those local minima, they turn out to be
>> converging on something astonishingly close to 9. For n_0 = 6 we get:
>> 
>> 8.69230769230769 (= 113/13)
>> 10.8230088495575
>> 9.15535568274734
>> 8.92837367151916
>> 8.90635284232427
>> 8.98815557904124
>> 8.99700731502177
>> 8.99853858951305
>> 8.99865861273323
>> 8.99952817610636
>> 
>> The ratios in other examples look similar, eg for 71:
>> 
>> 8.01910828025478
>> 8.90627482128674
>> 9.09542495317934
>> 9.09126653397002
>> 9.01309546792409
>> 9.00090632116533
>> 9.00247399195550
>> 9.00106389627424
>> 9.00051634900210
>> 
>> That makes no sense to me. I hope someone else can explain it.
>> 
>> Hugo van der Sanden
>> ---
>> 0 3
>> 1 2
>> 2 1
>> 3 6
>> 4 3
>> 5 2
>> 6 unknown
>> 7 4
>> 8 69
>> 9 6
>> 10 3
>> 11 58
>> 12 23
>> 13 10
>> 14 5
>> 15 12
>> 16 unknown
>> 17 4
>> 18 69
>> 19 6
>> 20 unknown
>> 21 8
>> 22 21
>> 23 56
>> 24 369019
>> 25 58
>> 26 23
>> 27 10
>> 28 5
>> 29 12
>> 30 unknown
>> 31 14
>> 32 7
>> 33 16
>> 34 37
>> 35 18
>> 36 9
>> 37 122
>> 38 11
>> 39 30
>> 40 69
>> 41 6
>> 42 unknown
>> 43 8
>> 44 21
>> 45 56
>> 46 369019
>> 47 58
>> 48 23
>> 49 10
>> 50 unknown
>> 51 unknown
>> 52 25
>> 53 70
>> 54 27
>> 55 12
>> 56 unknown
>> 57 14
>> 58 7
>> 59 16
>> 60 37
>> 61 18
>> 62 9
>> 63 122
>> 64 11
>> 65 30
>> 66 69
>> 67 8458
>> 68 13
>> 69 36
>> 70 unknown
>> 71 unknown
>> 72 15
>> 73 44
>> 74 107
>> 75 46
>> 76 unknown
>> 77 8
>> 78 21
>> 79 56
>> 80 369019
>> 81 58
>> 82 23
>> 83 10
>> 84 unknown
>> 85 unknown
>> 86 25
>> 87 70
>> 88 27
>> 89 12
>> 90 unknown
>> 91 14
>> 92 unknown
>> 93 4234
>> 94 33
>> 95 4336
>> 96 233
>> 97 16
>> 98 37
>> 99 18
>> 100 9
>> 
>> 
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>> 
> 
> --
> Seqfan Mailing list - http://list.seqfan.eu/