[seqfan] Re: Riecaman
hv at crypt.org
hv at crypt.org
Thu Sep 12 15:41:18 CEST 2019
Super stuff, could you please update the A-file with the result for 71
and the new limits for the incomplete ones?
Cheers,
Hugo
Nick Matteo <kundor at kundor.org> wrote:
: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
:
:--
:Seqfan Mailing list - http://list.seqfan.eu/
More information about the SeqFan
mailing list