Hey guys! I was recently introduced to these forums (and this post) and thought I would add an observation I had today.
We can say that the number of times it takes for an integer "n" to reach 1 is called the stopping time, or ns. I decided to represent these numbers as combinations of other stopping times + 1. For example, the first 10 would look like this:
1s = 0
2s = 1 = 1s + 1
3s = 7 = 10s + 1
4s = 2 = 2s + 1
5s = 5 = 16s + 1
6s = 8 = 3s + 1
7s = 16 = 22s + 1
8s = 3 = 4s + 1
9s = 19 = 28s + 1
10s= 6 = 5s + 1
What I noticed, is that if I continue writing the numbers out as combinations of other stopping times, each odd integer had a stopping time that paired with another even stopping time:
3s = 7 = 10s + 1
20s = 7 = 10s + 1
5s = 5 = 16s + 1
32s = 5 = 16s + 1
7s = 16 = 22s + 1
44s = 16 = 22s + 1
It may also be worth noting that the second number (higher one) increases at a constant rate. So we could write each pair as (ns , 6ns+2)
This is mostly just an observation, I haven't had too much time to actually see if this can be applied to get anywhere with the conjecture.
Hey guys! I was recently introduced to these forums (and this post) and thought I would add an observation I had today. We can say that the number of times it takes for an integer "n" to reach 1 is called the stopping time, or ns. I decided to represent these numbers as combinations of other stopping times + 1. For example, the first 10 would look like this: 1s = 0 2s = 1 = 1s + 1 3s = 7 = 10s + 1 4s = 2 = 2s + 1 5s = 5 = 16s + 1 6s = 8 = 3s + 1 7s = 16 = 22s + 1 8s = 3 = 4s + 1 9s = 19 = 28s + 1 10s= 6 = 5s + 1 What I noticed, is that if I continue writing the numbers out as combinations of other stopping times, each odd integer had a stopping time that paired with another even stopping time: 3s = 7 = 10s + 1 20s = 7 = 10s + 1 5s = 5 = 16s + 1 32s = 5 = 16s + 1 7s = 16 = 22s + 1 44s = 16 = 22s + 1 It may also be worth noting that the second number (higher one) increases at a constant rate. So we could write each pair as (ns , 6ns+2) This is mostly just an observation, I haven't had too much time to actually see if this can be applied to get anywhere with the conjecture.