Prime omega function

In number theory, the prime omega functions ${\displaystyle \omega (n)}$ and ${\displaystyle \Omega (n)}$ count the number of prime factors of a natural number ${\displaystyle n.}$ The number of distinct prime factors is assigned to ${\displaystyle \omega (n)}$ (little omega), while ${\displaystyle \Omega (n)}$ (big omega) counts the total number of prime factors with multiplicity (see arithmetic function). That is, if we have a prime factorization of ${\displaystyle n}$ of the form ${\displaystyle n=p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\cdots p_{k}^{\alpha _{k}}}$ for distinct primes ${\displaystyle p_{i}}$ (${\displaystyle 1\leq i\leq k}$), then the prime omega functions are given by ${\displaystyle \omega (n)=k}$ and ${\displaystyle \Omega (n)=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{k}}$. These prime-factor-counting functions have many important number theoretic relations.