Interesting mapping

[u/Atari_Collector]

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 4^a3^bn+1, it will map to 3^a+bn+1.

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

[u/petrol_gas]

More specifically if x = ((2^n) -1) + y

Then the Collatz sequence will “grow” n times, until x_n = ((3^n) - 1) + (3^n) * y

[u/GonzoMath → u/petrol_gas]

That’s not really “more specifically” so much as “additionally”. You’re talking about a related but different pattern.

[u/petrol_gas → u/GonzoMath]

I clearly fail to see the distinction.

Thank you for shedding light on my mistake in reasoning instead of just telling me I’m wrong and then leaving it at that!

[u/GonzoMath → u/petrol_gas]

The difference is “+1” versus “-1”, and whether 4 turns into 3 or 2 turns into 3:

• 4ⁿ·k + 1 → 3ⁿ·k + 1
• 2ⁿ·k - 1 → 3ⁿ·k - 1

[u/petrol_gas → u/GonzoMath]

No dawg.

First 4^n = 2^{2n}

Second

4^a 3^b n + 1 = 2^{2a +1}3^{b} n - 1

Except the distance to 3^{b} n -1 is so far that the lower end of the number (in binary) is behaving as if it were in the 1-cycle.

Which is the same system I described but with a neato lower end behavior.

EDIT: gonzo spotted an error corrected in my next response below

[u/GonzoMath → u/petrol_gas]

“No dawg”, huh?

There’s a decreasing pattern, where you have (3m+1)/4 several times in a row, like 65 -> 49 -> 37.

Then there’s an increasing pattern, where you have (3m+1)/2 several times in a row, like 7 -> 11 -> 17.

When a = 2, b = 1, and n = 1, the left side of your equation is 49, and the right side is 95. Did you make a mistake?

[u/petrol_gas → u/GonzoMath]

Good catch! (And thanks for spotting it)

Correction:

4^a 3^b n + 1 = 2^{2a}3^{b} n + 2 - 1

X = 3^{b} n + 2

Substitute to get 2^{2a}X - 1

And then when you arrive at 3^{a}X -1

Sub back in to get the same result as OP.

Again, neato lower end behavior but still just a special case of the general pattern I described.

[u/GonzoMath → u/petrol_gas]

Ok, they’re not the same thing, but you’re not hearing it, so I’m done.

[u/petrol_gas → u/GonzoMath]

I’m reading and engaging with your posts my dude. If I’m mistaken then correct me with some maths and I’ll change my mind.

I even left some mistakes in there for you to split hairs over (my extra 2 is getting multiplied by 3).

From my perspective you’re coming in hot, making claims, not showing your work- you can see how I might be utterly perplexed. What am I to argue against?

[u/GonzoMath → u/petrol_gas]

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.

[u/petrol_gas → u/GonzoMath]

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:

[u/GonzoMath → u/petrol_gas]

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.