Right now I'm calling a convergent number based on the Collatz conjecture the "Collatz constant". I'm wondering if it have an actual name? And if it actually converges?
Details
The Collatz conjecture provides an algorithm, namely, starting at any positive integer $i$ greater than 1, if $i\equiv 0 \pmod2$ then take $\frac{i}{2}$; otherwise take $3i+1$ and continue.
That gives us a sequence of numbers (assuming the sequence always comes back to $1$). So for $10$ we have $10, 5, 16, 8, 4, 2, 1$.
Consider adding one to each element in that sequence (ignoring the last element, 1) and taking their values $\pmod{2}$. That gives us $11, 6, 17, 9, 5, 3$ all $mod$ 2 which is $1, 0, 1, 1, 1, 1$.
Now interpret this as a binary number, least significant digit first. In which case $101111_2$ is $61$ ($111101$ in traditional nomenclature).
I call $61$ the Collatz number of $10$. Let's call the Collatz number function $C$.
Now consider the sequence $C(i+1)-C(i)$ for $i$ from 1 to $n$.
I maintain the ratio of primes (specifically elements whose absolute values are prime) to non-primes in that sequence converges to a real number as $n \rightarrow\infty$, what I'm calling the Collatz constant.
For $n = 10, 100, 1000, 10,\!000, 100,\!000$, and $1,\!000,\!000$ the approximation of the Collatz constant is:
10: 0.25
100: 0.515152
1,000: 0.466276
10,000: 0.462202
100,000: 0.461305
1,000,000: 0.45999
(I have a conjecture of my own about Collatz numbers. You can see it here. Drop me a PM if you find it interesting.)
I've been having trouble posting questions that are clear to readers, so I hope this is better.