In number theory, the prime omega functions ω ( n ) {\displaystyle \omega (n)} and Ω ( n ) {\displaystyle \Omega (n)} count the number of prime factors of a natural number n . {\displaystyle n.} Thereby ω ( n ) {\displaystyle \omega (n)} (little omega) counts each distinct prime factor, whereas the related function Ω ( n ) {\displaystyle \Omega (n)} (big omega) counts the total number of prime factors of n , {\displaystyle n,} honoring their multiplicity (see arithmetic function). For example, if we have a prime factorization of n {\displaystyle n} of the form n = p 1 α 1 p 2 α 2 ⋯ p k α k {\displaystyle n=p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\cdots p_{k}^{\alpha _{k}}} for distinct primes p i {\displaystyle p_{i}} ( 1 ≤ i ≤ k {\displaystyle 1\leq i\leq k} ), then the respective prime omega functions are given by ω ( n ) = k {\displaystyle \omega (n)=k} and Ω ( n ) = α 1 + α 2 + ⋯ + α k {\displaystyle \Omega (n)=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{k}} . These prime factor counting functions have many important number theoretic relations.