Prime omega function

In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, 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 <a href="/facts/Arithmetic_function/hULxwsgm">arithmetic function</a>). That is, if we have a <a href="/facts/Prime_factorization/EjMCTz2t">prime factorization</a> 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.

Prime omega function open-in-new

Prime omega function