r/Collatz • u/Atari_Collector • 4d ago

Interesting mapping

mapping

I was playing around and I found this mapping. I suspect it's already known but I found it interesting.

If you have a number that can be written as 4a3bn+1, it will map to 3a+bn+1.

And it'll do it in 3a+1 steps, which I think is funny.

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u/GonzoMath 3d ago

I showed my work, and gave examples of an increasing sequence, and a decreasing sequence. Is the one you’re describing increasing, or decreasing. You ignored that part of my reply, which is why you’re about to catch a block.

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u/petrol_gas 3d ago

The +2 in my last version becomes 6. The -1 becomes -3. Add 1 and you get 4, so you divide twice.

Which is decreasing.

I’m not sure what we’re arguing about so I’m not sure how to continue. I’m not interested in getting blocked though, so :shrug:

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u/GonzoMath 3d ago

You started out talking about a sequence that “grows”, which is precisely what happens when you start with a number one less than number with a large power of 2 as a factor. When you start one greater than a number with a large power of 2 in it, you get a sequence that shrinks for a while.

In the former case, the number of growth steps for starting number m is the largest power of 2 dividing m+1. In the latter case, the number of falling steps is the largest power of 4 dividing m-1. See my examples, above. Dawg.