3 Major results on \(\Omega \)’s
Let \(n\) and \(a\) be integers such that \(n{\gt}1\) and \(\left| a\right| {\gt}1\). If \(p\) is a prime number dividing \(\Omega _{n}(a)\) then there exists a nonnegative integer \(i\) such that \(n=\text{Che}_{p}(a)p^{i}\). If \(i{\gt}0\), then \(p\) is the greatest prime divisor of \(n\). If moreover \(p^2 \mid \Omega _{n}(a)\) then either \(p=2\) and \(n\in \left\lbrace 2,4\right\rbrace \), or \(p=3\) and \(n\in \left\lbrace 3,6\right\rbrace \).
Let \(n\) and \(a\) be integers such that \(n{\gt}1\) and \(\left| a\right| {\gt}1\). The following statements are equivalent.
A prime number \(p\) such that \(n=\text{Che}_{p}(a)\) does not exist.
\(\left| \Omega _{n}(a)\right|\) is a power of a prime number dividing \(n\).
\(\left| \Omega _{n}(a)\right|\) is a power of the greatest prime divisor of \(n\).