Composite numbers and Lhermite

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

φ for composite numbers

$\mathrm{\forall }n\in {\mathbb{N}}^{\mathbb{*}}$

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

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

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

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

Expresion of (X_n) according to Lhermite's model

${𝕏}_n=\sum _{i=1}^{4m}(\left[\frac{1+\sum _{m=1}^i\phi \left(m\right)}{n+1}\right]\times \left[\frac{n+1}{1+\sum _{m=1}^i\phi \left(m\right)}\right]\times i\times \phi \left(i\right))$

${𝕏}_n=\sum _{i=1}^{2m+2}(\left[\frac{1+\sum _{m=1}^i\phi \left(m\right)}{n+1}\right]\times \left[\frac{n+1}{1+\sum _{m=1}^i\phi \left(m\right)}\right]\times i\times \phi \left(i\right))$