Composite numbers and Lhermite

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

φ 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 (X_n) according to Lhermite's model

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

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