[seqfan] Re: Riecaman
Nick Matteo
kundor at kundor.org
Tue Sep 10 19:27:40 CEST 2019
I computed more terms of some of these sequences.
Most notably, the trajectory for 71 ends after 158228253524 (ca. 158
billion) steps, the last prime subtracted being 4443679700533.
I computed a few more of the minima in your sequence starting with 6:
i n_i p_{i+1}
2 1 5
5 2 13
12 1 41
29 8 113
78 3 401
199 4 1223
508 1 3637
1355 2 11197
3592 1 33569
9589 8 99971
25752 1 296753
70579 30 890377
194228 5 2664229
539961 60 8002847
1507602 1 24007873
4228745 4 72001673
11913940 7 215991203
33690443 38 647909833
95581182 25 1943478491
272003821 16 5830319399
776082524 19 17489762779
2219823175 16 52470123707
6363074656 9 157409115779
18275879639 18 472222208149
52587587974 3 1416658480733
151572062973 22 4249982656133
437548314952 61 12749957904521
1264892207811 8 38249838428717
3661490934914 3 114749512555373
10612045942623 8 344248576871219
30792215692360 13 1032745848371989
Thus it couldn't hit 0 until around 90 trillion steps.
The first few minima starting with 20 are:
i n_i p_{i+1}
6 1 17
15 4 53
46 1 211
113 10 619
280 5 1823
731 6 5531
1894 9 16339
5007 38 48679
13492 1 145819
36887 10 439367
101126 5 1315661
280177 18 3954053
779238 7 11857073
2179061 30 35561137
6122110 21 106653377
17273371 14 319963883
48904804 15 959782393
138917293 12 2879370899
395717222 87 8638014793
1130187981 4 25915353869
3235199798 49 77745289207
9280360825 36 233232717407
26672973902 7 699697233679
76797750557 32 2099101861691
221477503948 31 6297300594289
639681173681 58 18891882066913
1850131571990 81 56675642260597
The first few minima starting with 50 are:
i n_i p_{i+1}
10 1 31
23 6 89
64 1 313
159 2 941
398 5 2731
1037 12 8273
2754 1 24923
7299 8 73973
19506 1 218761
53301 26 656429
146396 9 1962967
405971 4 5892959
1132492 25 17685539
3173065 34 53053547
8928446 19 159130943
25226291 6 477406117
71506842 95 1432061549
203333407 12 4296147983
579758492 7 12887968169
1657251341 24 38665453867
4747662354 23 115994882567
13628787329 54 347979525269
39196613838 9 1043932282319
112924461059 38 3131800435073
325846539826 13 9395396532143
941613240105 10 28186159828283
When starting with 51, the minima occur at the same steps, but each
n_i is one larger.
For 70:
i n_i p_{i+1}
13 2 43
36 1 157
83 4 433
204 13 1259
531 8 3833
1356 15 11213
3611 8 33739
9764 5 101987
26731 10 309109
73274 15 927191
202605 2 2788813
562190 3 8356861
1569627 20 25062967
4406454 29 75219323
12419599 22 225710143
35130326 1 677159999
99702241 8 2031695719
283769884 7 6095160419
809844705 10 18286822013
2316543714 53 54859591001
6640906387 22 164578508771
19075689854 35 493739715163
54893524725 2 1481228734481
158228253522 19 4443679700513
456789722191 26 13331029653661
1320592376350 1 39993122267353
Similarly to the 50/51 pair, the trajectory for 71 has minima at the
same locations, but each n_i is one larger for the first 158 billion
steps.
When the trajectory for 70 hits 19 at step 158228253522, the
trajectory for 71 instead hits 20, which is equal to the next prime
difference 4443679700533 - 4443679700513,
so that the sequence terminates two steps later.
Cheers,
Nick Matteo
On Wed, Aug 28, 2019 at 6:46 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
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