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Old 2020-02-08, 13:48   #1
Boltzmann brain
Feb 2020

11 Posts
Default 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 = 112. 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:
I haven't tested these yet because I've been focusing on 22246.
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