# I call it the Germain function

$g(n)= \begin{cases} 1, \quad \quad\quad \quad\quad\quad\quad\quad\quad \quad n = 2^0, 2^1, 2^1, 2^2, \ldots \\ \displaystyle \prod_{\text{odd prime }p} \left( \dfrac{p-1}2 \right)^{v_p (n) }, \quad \text{otherwise} \end{cases}$ Let $$g(n)$$ be defined as above, where $$v_p(n)$$ denotes the highest power of prime $$p$$ that divides $$n$$.

Let $$M$$ be the set of all rational numbers $$\mu$$ for which there is at least one positive integer n, such that $$\mu=\log _{ n }{ g(n) }$$. In other words:

$$M=Q\cap \left\{ \log _{ n }{ g(n) } :\quad n\in \mathbb N \right\}$$,

wher $$Q$$ is the set of all rational numbers and $$N$$ the set of all positive integers.

What is the cardinality of $$M$$?

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