All Questions

Is there a formula to construct a Collatz (3x + 1) sequence of arbitrary length that is strictly increasing? Obviously one can do this with a strictly decreasing sequence by just taking $2^n$ but I ...

Let's call a sequence of nonnegative integers $x_1,x_2,\ldots$ matrix-realizable, if there exists a $k\times k$ nonnegative integer matrix $A$ (for some $k$), as well as nonnegative integer vectors $u,...

Let $$x_{n+1} = \begin{cases} x_n/2 &;\text{if } x_n \equiv 0 \pmod{2}\\ k\,x_n+1 &; \text{if } x_n\equiv 1 \pmod{2} \end{cases}$$ and $k=3$ and $x_n\in\Bbb N$. Collatz conjectured for this ...