Let's compute some of the sets $X_n = \{ x \in 2\Bbb{Z} +1 : f^{n}(x) = 2^i \text{ for some } i\geq 0\}$ where $f$ is the Collatz function. $$ f(x) = \dfrac{3x + 1}{2^{i(x)}} = 1 \iff (2^{i(x)} - 1) = 3x \iff 2^{i(x)} = 1 \pmod 3 $$ In this case each $i(x) \in 2\Bbb{N}$ will work and its correspo...

Is $n^2 + n$ always an even number?

The set of all $x\in \Bbb{Z}$ such that $\dfrac{3x + 1}{2^z} = 1$ for some $z \geq 0$ is closed under $x\cdot y := 3xy + x + y$. In fact this gives the set of iteration degree 1 Collatz solutions the structure of a monoid, with $0$ as the identity. For proof, just multiply out $1 = \dfrac{3x + ...