[seqfan] Re: Riecaman
Neil Sloane
njasloane at gmail.com
Thu Aug 29 04:29:25 CEST 2019
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
>
>
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