we show that all 2-free sequences end in a cycle of length 1. The 3-free Fibonacci sequences are much more complicated and we study them in Section \[sec:3free\]. All our computational experiments ended in a cycle of length 3. On the other hand, we show that 3-free sequences may contain arbitrary long increasing substrings. We prove this in Section \[sec:customized\]. Nevertheless, we give a probabilistic argument that a 3-free sequence should end in a cycle in Section \[sec:3free\].

The 4-free Fibonacci sequences are vastly different from 2-free and 3-free sequences (Section \[sec:4free\]). We did not find a sequence that ends in a cycle: all of them grow in our experiments. The proof that all of them grow seems intractable, but we supply a probabilistic argument that this is the case. Yet 5-free sequence bring something new (see Section \[sec:5free\]). They contain sequences that are never divided by 5 and provably grow indefinitely. At the same time 5-free sequences contain cycles too.

We continue with Section \[sec:division-free\] where we find other numbers $n$ that provide examples of sequences that never need to be divided by $n$. Now we wonder where the cycles disappeared to and discuss there potential properties in Section 