20200208, 13:48  #1 
Feb 2020
11 Posts 
Binary Complement Sequences and 22246
First post here, hopefully in the right place, I did look around a bit.
I have an interest in exceptions, outliers and holdouts. As an example, I have more interest in 27 having an unusually long collatz sequence, than I do in actually solving the problem. This is a particularly relevant example, since the sequences here are not too dissimilar. You probably know what the collatz sequences/hailstone numbers are, but I'll restate it anyway just in case: start with any positive integer. If odd, multiply by 3 and add 1. If even, divide by 2. Repeat this until it reaches 1 or loops. Simple enough. Most numbers return to 1 pretty rapidly, though there are a few numbers, such as 27 that take considerably longer. As I stated already, I'm interested in this sort of thing and have messed around with a few different rulesets, but that is for another thread, assuming there is interest in more daft sequences. Ok, I'll stop waffling now. I have called this "binary complement sequences" for reasons that will be obvious. Start with any positive integer. Multiply by 3, then find the binary complement. This is a single step. Repeat until it reaches 0 or loops. A binary complement is what it says on the tin  convert the number to binary, and turn every 1 into a 0, and every 0 into a 1. They 'complement' eachother, and sum to a mersenne number. The result can be anything from being 1 less than the term before, to actually being 1. Here is a concrete example: Our starting term is 2. multiplying this by 3 is equal to 6, which is 110 in binary. The complement is therefore 001, or 1. now, repeat with 1. 1*3 = 3 = 11_{2}. complement = 00 or 0 and the sequence has ended, with terms 2,1,0. Numbers generally reach 0 pretty quickly, though the exceptions to this are more numerous, and more extreme. 28 takes 7573 steps, reaching a maximum of the 54 digit number 123130640068522377168864228132316865867184046004226894. This isn't bettered until 227, which promptly nukes it with 664476 steps and 3.26*10^552. Note that when I say bettered, I mean excluding numbers that rapidly connect to the 28 sequence. 821 takes 3.18 million steps, but the reason I'm writing in this forum is the next record holder, 22246. It exceeds 10 million steps (@~10^1647), and going for 100M resulted in an 18 hour wait for a timeout error (I have only beginner knowledge of python and what I'm doing it on isn't designed for this sort of thing). Since I know you guys deal with huge numbers, I thought this would be the ideal place to ask for help on resolving it, and more if you are interested in this random sequence I came up with :3 Up to 30,000, the next numbers of interest (>100K steps) are: 22334 22630 24307 25412 25688 25856 26044 26251 26494 26710 27347 27423 28727 28813 I haven't tested these yet because I've been focusing on 22246. 
20200208, 19:03  #2  
Nov 2003
2^{2}·5·373 Posts 
Quote:
Do you have any (gasp!) mathematics to discuss? Or even probabilistic heuristics? This is just numerology. Have you read Terry Tao's blog on the subject? Or Jeff Lagarias' papers? Show us some math! 

20200208, 20:59  #3 
Feb 2020
11 Posts 
I wouldn't call it numerology, at least in the dictionary definition of the word. This to me is purely recreational mathematics. I think it is a bit unfair to suggest it isn't mathematics, though I'm not going to deny its lack of substance. If someone finds it has some relevance, I'd be surprised, but its not the goal. I was just messing around. Regarding the other things you mentioned, no I have neither read those, nor have any probabilistic heuristics. In truth, I don't have the mathematical ability to do that in the first place. Sorry for seemingly wasting your time.

20200209, 00:51  #4  
Nov 2003
2^{2}×5×373 Posts 
Quote:
making any kind of pronouncement about a new subject? (New, that is at least to you)?? This is a math subforum. Blind numerology isn't math. 

20200209, 01:33  #5 
Feb 2020
11 Posts 
I think you misunderstand me a little. I'm just an idiot with too much time on my hands having a bit of fun and wanting to share it. Not trying to make any pronouncements or say anything profound here at all! I have done background reading, but not to the technical level that I think would satisfy you enough to neutralise your criticism. Since you have restated this being numerology and not "maths", is there any subforum that *would* be suitable (in which case, would mods kindly move this to a more appropriate area). If not, am I (and this content) simply unwelcome here?

20200209, 02:36  #6 
"Matthew Anderson"
Dec 2010
Oregon, USA
2^{2}·149 Posts 
Here is an interesting article with an animated gif file regarding Collatz Conjecture.
Quanta magazine Also, see Online Encyclopedia Regards, Matt 
20200209, 03:04  #7  
Nov 2003
2^{2}·5·373 Posts 
Quote:
However, trying to discuss a subject in which you know you lack knowledge and then refusing to study the subject does make one an idiot. This subforum is for actual mathematics. There is the misc math subforum and the blogorhea for less than knowledgeable discussion. 

20200209, 03:34  #8 
Feb 2020
13_{8} Posts 
I had already read the Quanta article, and am very familiar with OEIS. I have read quite a bit about the collatz conjecture in fact, and have done so for 10+ years since learning it as a kid. I'm not quite as ignorant as I perhaps made myself out to be, but I am acutely aware that I do not have the higher mathematical knowledge to understand the analytics to anything more than a conceptual level (I dropped out of uni a few years back from an astrophysics course in 2nd year, so my maths knowledge is decent enough for anything outside of mathematical fields).
As for the sequence I posted, there are no loops up to at least 30,000. Trying 5 as a multiplier very likely leads most sequences to infinity as they grow rapidly and the chance of hitting a number sufficiently close to a mersenne number becomes vanishingly small too quickly. 
20200209, 13:43  #10 
Feb 2020
11 Posts 
I have been subscribed to Numberphile for several years, I do enjoy their content, especially some of the more esoteric stuff. As for the Collatz Conjecture itself, the answer is very likely yes everything does go to 1 eventually, though obviously the proof of that is well known to be elusive. I have messed around with alternate rules, such as: Divide by 2 or 3 if possible, else multiply. multiplier of 7 looks similar to the standard rule in that everything goes to 1, and 11 probably is infinite for most numbers. I went up to dividing by 2,3,5 and 7 and multiplying by 23, looking for rules that were as borderline between bounded and unbounded as possible. Interesting to me, though perhaps not very productive.
I did end up posting something else in misc maths as recommended by R.D. Silverman, so hopefully someone finds that interesting enough to engage. 
20200210, 11:12  #11 
Sep 2002
Vienna, Austria
219_{10} Posts 
Simple heuristic argument: the "binary complement" operation roughly divides a number by 4 on average (integrate log((1a)/(1+a)) on [0,1] gives 2log2), so most starting numbers "should" converge to 0.

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