The Syracuse map takes odd values

Let $T(n) = (3n+1)/2^{\,v_2(3n+1)}$ be the Syracuse map. Then $T(n)$ is odd for every $n$:

By construction $T(n)$ is the odd part of $3n+1$, that is, the quotient of $3n+1$ by the largest power of $2$ dividing it, so no factor of $2$ survives. Since $3n+1 \ne 0$ for every natural $n$, the statement needs no hypothesis.

The fact is what makes the Syracuse map a self-map of the odd numbers, and hence what allows its iterates to be formed and compared with the Collatz orbit.