Shrinking Numbers
Do we always get smaller numbers in our sequence?
Last updated
Do we always get smaller numbers in our sequence?
Last updated
We can analyze the table to determine at which point a cell value is an integer less than it's row number. What that means is that, for such a cell, using the trajectory defined by the column ECF and starting from the number of that row, we will reach a smaller number. We can then re-iterate this logic until we go all the way down to one, if possible.
When we look at the shrinking numbers, there is a pattern with respect to the powers of two.
It is trivial to see that number of the form are even, and thus they will immediately shrink within a single iteration of Collatz Function.
Numbers of the form shrink with ECF , bijectively mapped to 2.
We do not have an immediate proof for odd numbers though. So, let us split the odd numbers in two as and .
There is actually a neat proof that numbers of the form shrink with ECF .
Indeed, for positive integers .
Yeah, indeed this shrinks the same way!
We will use proof by induction. Let us see the first few results:
This gives the sequence 2, 5, 11, 23, 47, 95, 191, …; and is a perfect opportunity to see if OEIS has it! Indeed it has: OEIS A083329.
The ECFs follow the pattern:
I had to use Wolfram|Alpha to see that:
So, what we have is:
Numbers of the form shrink with ECF , bijectively mapped to 5.
Again, we don’t know what is the case for . So we go one more level beyond, and look at numbers and .
Candidates: 1 → 0, 2 → 0, 1, 3 (add 1 and prepend 0). Or, look at their bijective mappings: 2 → 5 → 11? Well, 11 would give ECF so that fits our previous guess!
Numbers of the form shrink with ECF , bijectively mapped to 11.
n | Numbers | ECF | Bijective Map |
---|---|---|---|
A general form for the numbers w.r.t can be written as . As for the ECFs, we can define them by the following recursion:
There is also OEIS A055010 but only the leading term is different, which we do not consider since we start with .
Let us write the ICF for some number for ECF .
That is cool and all, but sadly does not turn out to be an integer each time. When it does, it does show that the number is shrinking though!
1
2
2
5
3
11
4
23
5
47