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{\displaystyle \left(\mathbb {X} _{n}\right)}
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{\displaystyle n^{th}}
φ for composite numbers
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{\displaystyle \forall n\in \mathbb {N^{*}} }
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{\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} }
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{\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} }
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{\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]}
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{\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 (Xn ) according to Lhermite's model
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{\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)}}
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{\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)}}
![{\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} }](./12ccb6c88be7061641216b48bf2ec71e01c3aee6.svg)
![{\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} }](./cbdb236490012cd5b81fd7118593d9003a32ce17.svg)
![{\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]}](./d8690258e711c5a7303b37bcfcac4dae3035df64.svg)
![{\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)}}](./2d5195df0996d15b7ba9adb903b0100222d14241.svg)
![{\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)}}](./7f3dabf8d67c13833afd35ebe7a358ad6766340d.svg)
![{\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)}}](./5e9a4d52af6fcc795db7f974c2ea567b5cbba180.svg)