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Consider $$n \rightarrow ... \rightarrow an+b$$ to be a sequence of natural numbers to which we apply $3n+1$ or $\frac{n}{2}$ operations. This sequence is called a cycle, if and only if $an+b=2n$. (one can define it as $an+b=\frac{n-1}{3}$ as well, but it is the same thing, just shifted to other ...