[seqfan] Re: Riecaman

Thu Aug 29 14:07:43 CEST 2019

  This got me thinking about representing integers in the form +/- A0 +/-
A1 +/- A2 +/- ... +/- An.
For any sequence A, one can form a new sequence Bn, the length of the
shortest such
representation. For example, with A = A001477 (the natural numbers) the
first few shortest
representations are:

0 = (empty sum)
1 = 1
2 = 1 + 2 - 3
3 = 1 + 2
4 = - 1 + 2 + 3
5 = 1 - 2 - 3 + 4 + 5
6 = 1 + 2 + 3
7 = 1 + 2 + 3 - 4 + 5

So the sequence derived as I propose from A001477 begins 0, 1, 3, 2, 3, 5,
3 ...

This is in OEIS at A140358, except that A140358 lacks the initial 0 for
reasons best known to the author.

So what about the corresponding sequence for A000040, the primes?

0 = 2 + 3 - 5
1 = - 2 + 3
2 = 2
3 = - 2 + 3 - 5 + 7
4 = 2 - 3 + 5
5 = 2 + 3
6 = - 2 + 3 + 5
7 = 2 + 3 - 5 + 7
8 = 2 - 3 + 5 - 7 + 11
9 = 2 - 3 + 5 + 7 + 11 - 13
10 = 2 + 3 + 5

The sequence derived in this way from the primes thus begins 3, 2, 1, 4, 3,
2, 3, 4, 5, 6, 3, and this is
not in OEIS, even without A0 = 3. I can glimpse a possible connection to
Goldbach's Conjecture here.

On Thu, Aug 29, 2019 at 7:45 AM <hv at crypt.org> wrote:

> :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.
> :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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> :>
> :
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