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avatar+810 

There are a couple rules to this problem.

1. If the number is even, please divide by 2.
2. If the number is odd, please multiply by 3 and add 1.

Now, let's have an example. 6.
6 is even, so we divide by 2 to get 3. 3 is odd so we have to multiply it by 2 and add 1 to get 10. 10 divided by 2 is 5. 5 times 3 plus 1 is 16. 16 divided by 2 is 8, 8 divided by 2 is 4, 4 divided by 2 is 2, and 2 divided by 2 is 1. 1 times 3 plus 1 is 4, 4 divided by 2 is 2, and 2 divided by 2 is 1.
As you can see, we are in a loop. Let's try another number. 7.

7 times 3 plus 1 is 22, divided by 2 is 11, times 3 plus 1 is 34, divided by 2 is 17.
I will list it out.

7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1.
As you can see, we are in the loop again.

Let's try a last number. 9.
9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1.

 

Is there a number that does no go through this loop?
 

NOTICE: THIS IS NOT COLLATZ. THERE ARE SOLUTIONS!!!

 May 23, 2024
 #1
avatar+1654 
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Yes, there are numbers that do not go through the loop in the described sequence of rules. This sequence defines a function that iterates on a number based on some rules. The loop you observed happens because the function eventually reaches numbers that have already been seen in the sequence.

Here's why some numbers might not enter the loop:

 

Odd numbers greater than 1: The function multiplies odd numbers by 3 and adds 1. If the resulting number is even (divisible by 2), the loop continues. However, if the resulting number is odd and greater than 1, the function will repeat the multiplication by 3 and addition of 1. This can potentially lead the sequence to entirely new, unseen numbers that haven't been encountered before.

 

Numbers divisible by 3 but not by 6: When a number is divided by 2, it might reach a point where the result is divisible by 3. If this number is divisible by 3 but not by 6 (meaning it has a remainder of 3 when divided by 6), the function will reach a number that is odd and greater than 1 (as described in point 1). This can again lead the sequence to explore new parts that haven't been seen before.

 

For example, the number 10 never enters the loop you observed. Here's why:

 

10 -> 5 (even, divided by 2) 5 -> 16 (even, divided by 2) 16 -> 8 (even, divided by 2) 8 -> 4 (even, divided by 2) 4 -> 2 (even, divided by 2) 2 -> 1 (odd, less than or equal to 1) - The sequence reaches 1, which terminates the process.

 

In conclusion, while the loop you observed occurs for many numbers, there are indeed numbers that the function explores and never enters the repetitive cycle. These numbers can fall under the categories mentioned above (odd numbers greater than 1 and certain numbers divisible by 3 but not by 6).

 May 23, 2024

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