Composite numbers and Lhermite

$$\left(\mathbb{X}_n\right)$$ is the $$ n^{th}$$ composite number.

φ for composite numbers
$$\forall n \in \mathbb{N^*} $$


 * $$1-\left[ \frac{\left[\frac{\left(n-1\right)!+1}{n}\right]}{\frac{\left(n-1\right)!+1}{n}}\right]=1 \Longleftrightarrow n \in \mathbb{X}$$

$$1-\left[ \frac{\left[\frac{\left(n-1\right)!+1}{n}\right]}{\frac{\left(n-1\right)!+1}{n}}\right]=0\Longleftrightarrow n \notin \mathbb{X}$$

$$\varphi\left(n\right)=1-\left[ \frac{\left[\frac{\left(n-1\right)!+1}{n}\right]}{\frac{\left(n-1\right)!+1}{n}}\right] $$

$$\varphi\left(n\right)={\left(\left[\frac{\left[\frac{\left(n!\right)^2}{n^3}\right]}{\frac{\left(n!\right)^2}{n^3}}\right]-\left[\frac{1}{n}\right]\right)} $$

Expresion of (Xn) according to Lhermite's model
$$\mathbb{X}_n=\sum_{i=1}^{4m}{\left(\left[\frac{1+\sum_{m=1}^{i}{\varphi\left(m\right)}}{n+1}\right]\times\left[\frac{n+1}{1+\sum_{m=1}^{i}{\varphi\left(m\right) }}\right]\times i\times\varphi\left(i\right)\right)}$$

$$\mathbb{X}_n=\sum_{i=1}^{2m+2}{\left(\left[\frac{1+\sum_{m=1}^{i}{\varphi\left(m\right)}}{n+1}\right]\times\left[\frac{n+1}{1+\sum_{m=1}^{i}{\varphi\left(m\right) }}\right]\times i\times\varphi\left(i\right)\right)}$$