I have just recently learned about the Collatz conjecture and have been researching it for fun.
For this question, $m$ and $N$ are natural numbers.
I am interested in $2^m$ that the natural number $N$ can reach by the Collatz Conjecture, so I am looking it up. Then I found A054646 in oeis.
This is a list of the smallest natural numbers that can reach $2^m$ by the Collatz Conjecture.
This sequence is monotonically increasing as far as it is currently computed.
This surprised me because my intuition was that the $2^m$ that could be reached for a natural number $N$ was random.
If this sequence is still monotonically increasing after this, then there is an upper limit of $2^m$ that can be reached for the natural number $N$.
Please let me know if you have any information for this sequence.
Concrete examples
$\frac{4^n-1}{3}$ is the odd number just before reaching $2^m$ ($n$ is natural number).
The ratio of $\frac{4^n-1}{3}$ to $A054646(n)$ is currently less than 4.
I suspect this is because there is an upper limit of $2^m$ that the natural number $N$ can reach.
| $\frac{4^n-1}{3}$ | $A054646(n)$ | $\frac{4^n-1}{3} / A054646(n)$ |
|---|---|---|
| 1 | 1 | 1.0 |
| 5 | 3 | 1.6666666666666667 |
| 21 | 21 | 1.0 |
| 85 | 75 | 1.1333333333333333 |
| 341 | 151 | 2.2582781456953644 |
| 1365 | 1365 | 1.0 |
| 5461 | 5461 | 1.0 |
| 21845 | 14563 | 1.500034333585113 |
| 87381 | 87381 | 1.0 |
| 349525 | 184111 | 1.8984471324364107 |
| 1398101 | 932067 | 1.5000005364421227 |
| 5592405 | 5592405 | 1.0 |
| 22369621 | 13256071 | 1.6875000895815961 |
| 89478485 | 26512143 | 3.375000089581593 |
| 357913941 | 357913941 | 1.0 |
| 1431655765 | 1431655765 | 1.0 |
| 5726623061 | 3817748707 | 1.5000000001309672 |
| 22906492245 | 22906492245 | 1.0 |
| 91625968981 | 91625968981 | 1.0 |
| 366503875925 | 244335917283 | 1.5000000000020464 |
| 1466015503701 | 1466015503701 | 1.0 |
| 5864062014805 | 5212499568715 | 1.12500000000012 |
| 23456248059221 | 10424999137431 | 2.25000000000012 |
| 93824992236885 | 93824992236885 | 1.0 |
More concrete examples
The Collatz sequence for the natural number $7$ is as follows.
$$ 7 \rightarrow 22 \rightarrow 11 \rightarrow 34 \rightarrow 17 \rightarrow 52 \rightarrow 26 \rightarrow 13 \rightarrow 40 \rightarrow 20 \rightarrow 10 \rightarrow 5 \rightarrow 16 \rightarrow 8 \rightarrow 4 \rightarrow 2 \rightarrow 1$$
My expectation in this question is that the reason why large powers of $2$, e.g., $2^{10}$, do not appear in this sequence is because $N$ is as small as $7$.