# [seqfan] Re: Riecaman

hv at crypt.org hv at crypt.org
Thu Aug 29 12:04:14 CEST 2019

```:Could you create sequences for the "primitive" trajectories 6 20 50 51 70
:71 say?
:With good fat b-files?

Sure. Is a million terms enough? I feel there's very little of interest
happening other than at the local minima though, so doing all 6 may be
overkill.

I want to submit the main sequence first though, but I've been searching
for information on formatting for an A-file (since there are gaps), and
have not managed to find any. It would be a shame to submit nothing but
the first 6 terms. Would it be simply a b-file with expressions for
unknowns, like this?

...
5 2
6 > 15000000000
7 4
...
16 = a(6) > 15000000000

:So where does your factor of 9 come from?

This turns out to be more straightforward than I had imagined: after an
even minimum reached at k, we reset to the low end being odd and in the
region of p_k; this is then reduced by every other prime difference, so
by the time we reach 3p_k, we're in the vicinity of 0. We then reset to
a low end that's even and in the region of 3p_k, so by the time we reach
9p_k, we're in the vicinity of 0 again.

I think the main remaining questions of interest are the probability that
a(n) = 0, and the expected magnitude of a(n) if non-zero, if we haven't
yet terminated on reaching p_k.

I suspect that the probability is 0, on the basis that the probability
of terminating at any given minimum is in the region of 1/ln(p_k), so we
get probabilities at successive minima of 1/ln(9^i p_k), essentially
a harmonic progression. I don't know how to estimate the expected
magnitude though: I'm expecting a scary answer.

Hugo

Neil Sloane <njasloane at gmail.com> wrote:
: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.
: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/
:>
:
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:Seqfan Mailing list - http://list.seqfan.eu/

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