I'm working on a function $\lambda(\xi)$ that takes in input an intger $\xi$ and calculate for how many values of $x\in N$, with $0\leq x < \xi$, the ratio: $$\psi = \frac{\xi x}{\xi -x}$$ is an integer $(\psi \in N)$.
Here there is a graph of $\lambda(\xi)$ for $2\leq \xi \leq113$, created in Excel:
I have tried in a lot of diferent ways (supposing for example $\xi$ is odd and $x$ is even), but I can't faind a solution. Any idea?
To prove the assertion of Peter Foreman in its comment, that's $$\lambda(\xi)=\frac{d(\xi^2)+1}2$$ let consider the sets \begin{align*} D&=\{d>0:d|\xi^2\}\\ L&=\{d\in D:d\leq\xi\}\\ U&=\{d\in D:d>\xi\} \end{align*} and the function \begin{align*} &\varphi:D\to D&&d\mapsto\xi^2/d \end{align*} Then \begin{align} D&=L\cup U& L\cap U&=\varnothing\\ &\text{$\varphi$ is bijective}&\varphi[U]&=L-\{\xi\} \end{align} Consequently, $d(\xi^2)=|D|=|L|+|U|$ and $|U|=|L|-1$ from which $$\lambda(\xi)=|L|=\frac{|D|+1}2=\frac{d(\xi^2)+1}2$$ thus proving the assertion.