[seqfan] Re: Riecaman
hv at crypt.org
hv at crypt.org
Sat Aug 31 15:26:48 CEST 2019
Hi Jens,
I quite agree with your first point, as laid out in my followup to NJAS.
Re your 4th point: in general, between one minimum and the next, the
trajectory goes from some value k just after the first minimum, reducing
every two steps by the difference of the next pair of entries in the
co-sequence, until it reaches (nearly) zero.
If the co-sequence is regular enough, or random enough, then those
differences will be close to half of the total difference; and if the
minima are small enough relative to k, then this means we'll be using
co-sequence entries from about k to about 3k.
Try a co-sequence such as { k: k^2 == 1 (mod 3) }, and I expect you'll
see a different ratio.
Re the 3rd point: this will be essentially the same - instead of going
from 0 + k down to 0, we're going from T + k down to T.
Hugo
jens at voss-ahrensburg.de wrote:
: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/
:
:
:--
:Seqfan Mailing list - http://list.seqfan.eu/
More information about the SeqFan
mailing list