From Surf Wiki (app.surf) — the open knowledge base

Prime omega function

Quality 0.00 · 2 views · Updated 1 month ago

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} . The number of distinct prime factors is assigned to ω ( n ) {\displaystyle \omega (n)} (little omega), while Ω ( n ) {\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 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 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.