Possible variation of the $"3n+1"$ problem as $"3n+K"$

Question

Variations of the Collatz problem

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Let

$$ f(n) = \begin{cases} 3n+K & \text {$n$ odd} \\ \frac{n}{2} & \text {$n$ even} \end{cases} $$ with $$K\in\mathbb Z^*$$

I am looking for $K\neq1$ such that $(\forall n\in\mathbb{N})\,\,\, f^k(n)=1$ for large enough $k$.

Answer 1

If $K\neq 1$ and it's odd, your sequence will always contain a cycle: $$K, 4K, 2K, K$$

If $K$ is even, one can see the sequence increases infinitely once it hits odd number since it never comes back into even number.