( X n ) {\displaystyle \left(\mathbb {X} _{n}\right)} is the n t h {\displaystyle n^{th}} composite number.
∀ n ∈ N ∗ {\displaystyle \forall n\in \mathbb {N^{*}} }
1 − [ [ ( n − 1 ) ! + 1 n ] ( n − 1 ) ! + 1 n ] = 0 ⟺ n ∉ 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} }
φ ( n ) = 1 − [ [ ( n − 1 ) ! + 1 n ] ( n − 1 ) ! + 1 n ] {\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]}
φ ( n ) = ( [ [ ( n ! ) 2 n 3 ] ( n ! ) 2 n 3 ] − [ 1 n ] ) {\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)}}
X n = ∑ i = 1 4 m ( [ 1 + ∑ m = 1 i φ ( m ) n + 1 ] × [ n + 1 1 + ∑ m = 1 i φ ( m ) ] × i × φ ( i ) ) {\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)}}
X n = ∑ i = 1 2 m + 2 ( [ 1 + ∑ m = 1 i φ ( m ) n + 1 ] × [ n + 1 1 + ∑ m = 1 i φ ( m ) ] × i × φ ( i ) ) {\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)}}
is the 
composite number.
![{\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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/12ccb6c88be7061641216b48bf2ec71e01c3aee6)
![{\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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbdb236490012cd5b81fd7118593d9003a32ce17)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8690258e711c5a7303b37bcfcac4dae3035df64)
![{\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)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d5195df0996d15b7ba9adb903b0100222d14241)
![{\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)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f3dabf8d67c13833afd35ebe7a358ad6766340d)
![{\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)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e9a4d52af6fcc795db7f974c2ea567b5cbba180)
