(Xn){\displaystyle \left(\mathbb {X} _{n}\right)} is the nth{\displaystyle n^{th}} composite number.
∀n∈N∗{\displaystyle \forall n\in \mathbb {N^{*}} }
1−[[(n−1)!+1n](n−1)!+1n]=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)!+1n](n−1)!+1n]{\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!)2n3](n!)2n3]−[1n]){\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)}}
Xn=∑i=14m([1+∑m=1iφ(m)n+1]×[n+11+∑m=1iφ(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)}}
Xn=∑i=12m+2([1+∑m=1iφ(m)n+1]×[n+11+∑m=1iφ(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)}}