This paper describes a class of sequences that are in many ways similar to Fibonacci sequences: given n, sum the previous two terms and divide them by the largest possible power of n. The behavior of such sequences depends on n. We analyze these sequences for small n: 2, 3, 4, and 5. Surprisingly, these behaviors are very different. We also present theorems regarding any n. Many statements about these sequences may be difficult or even impossible to prove, but they can be supported by probabilistic arguments. We have plenty of those in this paper.

We also introduce ten new sequences. Most of the new sequences are also related to Fibonacci numbers proper, not just free Fibonacci numbers.