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.
Properties and relations The function ω ( n ) {\displaystyle \omega (n)} is additive and Ω ( n ) {\displaystyle \Omega (n)} is completely additive. Little omega has the formula
ω ( n ) = ∑ p ∣ n 1 , {\displaystyle \omega (n)=\sum _{p\mid n}1,} where notation p|n indicates that the sum is taken over all primes p that divide n, without multiplicity. For example, ω ( 12 ) = ω ( 2 2 3 ) = 2 {\displaystyle \omega (12)=\omega (2^{2}3)=2} . Big omega has the formulas
Ω ( n ) = ∑ p α ∣ n 1 = ∑ p α ∥ n α . {\displaystyle \Omega (n)=\sum _{p^{\alpha }\mid n}1=\sum _{p^{\alpha }\parallel n}\alpha .}
The notation pα|n indicates that the sum is taken over all prime powers pα that divide n, while pα||n indicates that the sum is taken over all prime powers pα that divide n and such that n / pα is coprime to pα. For example, Ω ( 12 ) = Ω ( 2 2 3 1 ) = 3 {\displaystyle \Omega (12)=\Omega (2^{2}3^{1})=3} . The omegas are related by the inequalities ω(n) ≤ Ω(n) and 2ω(n) ≤ d(n) ≤ 2Ω(n), where d(n) is the divisor-counting function. If Ω(n) = ω(n), then n is squarefree and related to the Möbius function by
μ ( n ) = ( − 1 ) ω ( n ) = ( − 1 ) Ω ( n ) . {\displaystyle \mu (n)=(-1)^{\omega (n)}=(-1)^{\Omega (n)}.}
If ω ( n ) = 1 {\displaystyle \omega (n)=1} then n {\displaystyle n} is a prime power, and if Ω ( n ) = 1 {\displaystyle \Omega (n)=1} then n {\displaystyle n} is prime. An asymptotic series for the average order of ω ( n ) {\displaystyle \omega (n)} is
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