# Searching information on a certain function with a fixed point property connecting Moebius $\mu$ and Fibonacci numbers

Let $\mu$ be the Moebius function and define for $1\leq n\in\mathbb{N}$

$$f(n) = \left\{ \begin{array}{ll} \mu\left(\frac{n}{2}\right) + \mu\left(\frac{n}{4}\right), & n\equiv 0, 4, 8\mod 12, \\ 0, & n\equiv 2, 10\mod 12, \\ \mu(n) + \mu\left(\frac{n}{3}\right), & n\equiv 3, 9\mod 12, \\ \mu\left(\frac{n}{3}\right), & n\equiv 6\mod 12, \\ \mu(n), & n\equiv 1, 5, 7, 11\mod 12. \end{array} \right.$$ The first values of this function are $$1, 0, 0, 0, -1, -1, -1, -1, -1, 0, -1, 0, -1, 0, 0, 0, -1, 1, -1, 0, 0, 0, -1, 1, \ldots$$ and it satifies the following fixed point equation $$\sum_{\alpha(d)=n}f(d) = f(n)$$ where $\alpha(d)$ is the first index $n$ such that $d$ divides the Fibonacci number with index $n$ and no Fibonacci number with smaller index. Sometimes $\alpha$ is called the rank of apparition.

Q: Has this function been investigated? Maybe in a different context? If possible, please provide a reference.

OEIS did not help.

Motivation: Sums similar to the above can e.g be found in this question of mine on SE https://math.stackexchange.com/questions/1151893/connection-between-eulers-totient-function-and-fibonacci-numbers or in this question of mine on MO Sum of the log of all primes dividing at least one Fibonacci number up to index x