[seqfan] Re: Binary Complement Sequences

Wed Dec 28 19:37:07 CET 2022

349525 = 0x55555 and 699050 = 0xAAAAA, so they go to 0 after one iteration. Numbers slightly smaller give a small number after one iteration, while numbers slightly larger give a big number. It is not surprising that there are vertical lines there.

> On December 27, 2022 at 10:16 PM Tom Duff <eigenvectors at gmail.com> wrote:
> 
> 
> I have posted a plot of the # of steps for each starting number from 1 to
> 820559 to reach zero at http://iq0.com/fate.png
> (Doesn't include 717657 and 819991, which are still running, as is 820560,
> with those having run 19.9B, 5.2B and 6.9B steps so far. Maybe they'll be
> done in a week or so.)
> Note that the y axis of the plot is logarithmic.
> The horizontal lines are the most obvious feature. Presumably they're
> mostly numbers that quickly converge
> to a common trajectory -- I haven't checked.
> But the vertical stripes are the interesting part. I have no idea what's
> going on there.
> 
> Also, I posted a plot of iteration number vs iterate size (in bits) at
> http://iq0.com/iterations-vs-size.png that shows the progress of 425720,
> 819991, 717657 and 820560, plotted over top of one another in different
> colors. It's pretty clear that 819991 is following the trajectory of 425720
> and 820560 is following 717657. There's a legend in the upper right corner
> indicating the colors of each curve.
> 
> On Mon, Dec 26, 2022 at 10:27 AM Tom Duff <eigenvectors at gmail.com> wrote:
> 
> > I didn't expect this. I really thought it would diverge. This seriously
> > indicates that it invariably converges to zero. That, not the computation
> > of more values, is the front on which we need progress, now.
> >
> > On Mon, Dec 26, 2022 at 10:25 AM Tom Duff <eigenvectors at gmail.com> wrote:
> >
> >> And it just finished! 425720 takes 87,037,147,316 steps to converge to 0.
> >> (Or my  computer glitched, or I have a bug. I seriously doubt the latter,
> >> because all my other results match what others have reported.)
> >>
> >> On Mon, Dec 26, 2022 at 8:23 AM Tom Duff <eigenvectors at gmail.com> wrote:
> >>
> >>> My run of 425720 has been going for almost 83 billion iterations. The
> >>> length of the current iterate is down to under 167000 bits (from a maximum
> >>> of roughly 595000 bits). Excitement reigns!
> >>>
> >>> On Fri, Dec 16, 2022 at 11:00 AM Joshua Searle (larry) <
> >>> jprsearle at gmail.com> wrote:
> >>>
> >>>> Hello,
> >>>>
> >>>> (In my enthusiasm, I sent this first time around before I got
> >>>> confirmation of being added to the mailing list so I don’t think anyone saw
> >>>> it, oops)
> >>>>
> >>>> I am looking for some help finding some more terms for a set of
> >>>> sequences I intend to add to the OEIS.
> >>>>
> >>>> It is a similar algorithm to that of the collatz algorithm, but instead
> >>>> of of multiplying by 3 and adding when odd, and dividing when even, it goes
> >>>> as follows:
> >>>>
> >>>> on any number:
> >>>> -multiply by 3
> >>>> -find the binary complement (if it is 1001010 in binary, the complement
> >>>> is 0110101). This is equivalent to subtracting from the next highest
> >>>> mersenne number.
> >>>>
> >>>> this is treated as all one step, so a seed of 2 produces the sequence
> >>>> [2,1,0]
> >>>> 3 produces the longer [3, 6, 13, 24, 55, 90, 241, 300, 123, 142, 85, 0].
> >>>>
> >>>> For lack of a better name I’ve called these binary complement sequences.
> >>>>
> >>>> While you might expect similar behaviour to the collatz algorithm (and
> >>>> it largely does), it turns out this can support sequences that are
> >>>> staggeringly long in length. The starting seed of 28 takes 7572 terms to
> >>>> terminate and I terminated my code after seed 425720 exceeded 10 billion
> >>>> terms! I do think all sequences terminate.
> >>>>
> >>>> The following sequences can be made from it:
> >>>>
> >>>> 1a) step length: (seed = term 0, natural numbers)
> >>>> 1 <= n <= 30
> >>>> 1, 2, 11, 12, 1, 10, 3, 4, 13, 2, 19, 80, 9, 2, 15, 16, 81, 14, 11, 12,
> >>>> 1, 6, 83, 8, 73, 22, 79, 7572, 5, 18…
> >>>>
> >>>> 1b) max value: (natural numbers)
> >>>> 1 <= n <= 20
> >>>> 1, 2, 300, 300, 5, 300, 10, 10, 300, 10, 300, 328536, 300, 21, 300,
> >>>> 300, 328536, 300, 300, 300…
> >>>>
> >>>> 2a) seeds with record step length:
> >>>> 1 <= n <= 25, all known terms.
> >>>> 1, 2, 3, 4, 9, 11, 12, 17, 23, 28, 33, 74, 86, 180, 227, 350, 821,
> >>>> 3822, 4187, 5561, 6380, 6398, 22174, 22246, 26494
> >>>>
> >>>> 2b) step lengths of 2a:
> >>>> 1 <= n <= 25, all known terms
> >>>> 1, 2, 11, 12, 13, 19, 80, 81, 83, 7572, 7573, 7574, 7578, 7580, 664475,
> >>>> 664882, 3180929, 3180930, 3180931, 3181981, 3181988, 3182002, 3182226,
> >>>> 120796790, 556068798
> >>>>
> >>>> 2c) max values of 2a:
> >>>> 1 <= n <= 25, al known terms, abbreviated for readability
> >>>> 1, 2, 300 (x4), 328536 (x3), ~1.23*10^53 (x5), ~3.26*10^552 (x2),
> >>>> ~2.03*10^933 (x7), ~9.38*10^8306, ~1.67*10^16667
> >>>>
> >>>> 3a) seeds with record step length and new maxima (excludes all the side
> >>>> sequences, new maxima are not necessarily larger than the previous):
> >>>> 1 <= n <= 12, all known terms
> >>>> 1, 2, 3, 12, 28, 227, 821, 22246, 26494, 103721, 204953, 425720
> >>>>
> >>>> 3b) step lengths of 3a
> >>>> 1 <= n <= 11, all known terms plus a lower bound for next one.
> >>>> 1, 2, 11, 80, 7572, 664475, 3180929, 120796790, 556068798, 572086533,
> >>>> 1246707529, 9999999999+
> >>>>
> >>>> 3c) max values of 3a
> >>>> 1 <= n <= 11, all known terms plus a lower bound for next one.
> >>>> 1, 2, 300 , 328536, ~1.23*10^53, ~3.26*10^552, ~2.03*10^933,
> >>>> ~9.38*10^8306, ~1.67*10^16667, ~2.42*10^14081, ~9.81*10^25580,
> >>>> >=2.09*10^114778
> >>>>
> >>>> Observations and questions:
> >>>> -The max value achieved by a sequence has roughly sqrt(step count)
> >>>> digits.
> >>>> -For how many terms can a sequence continually increase? I haven’t
> >>>> tracked it but even 3 has 6 consecutively increasing terms in its sequence.
> >>>> -The penultimate term of a sequence must be of the form [(2^3n-1)-1]/3.
> >>>> I haven’t tracked how often sequences fall into these.
> >>>> -What does a log plot look like of these sequences? They have had far
> >>>> too many data points for basic graphing software to handle!
> >>>> -And of course, does every sequence terminate? (probably unanswerable)
> >>>>
> >>>> Being able to terminate 425720 would be nice, despite several drastic
> >>>> speedups from my rickety initial coding effort, still took 67 hours to
> >>>> compute 10 billion terms of the sequence. I can provide a data file where I
> >>>> copy and pasted results from general searches if requested. For example, I
> >>>> can give you term 9,999,999,999 of seed 425720, or the step lengths/maxima
> >>>> of sequences up to 425720 that didn’t get caught by my side-sequence filter.
> >>>>
> >>>> I’m worrying that this is too long; I hope that at least someone reads
> >>>> until the end!
> >>>>
> >>>> Joshua Searle.
> >>>>
> >>>> Email: jprsearle at gmail.com <mailto:jprsearle at gmail.com> (if you want
> >>>> to request files)
> >>>>
> >>>> --
> >>>> Seqfan Mailing list - http://list.seqfan.eu/
> >>>>
> >>>
> 
> --
> Seqfan Mailing list - http://list.seqfan.eu/