# Composite numbers and Lhermite

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


## ${\displaystyle \varphi }$ for composite numbers

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

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

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

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

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

## Expresion of ${\displaystyle \left(\mathbb {X} _{n}\right)}$ according to Lhermite's model

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

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