Math 2.0

Number theory Definition

Number of prime factors counted with multiplicity

Builds on Prime factorizationArithmetic function

Definition

Let nn be a positive integer with prime factorization n=p1e1⋯pkekn = p_1^{e_1} \cdots p_k^{e_k}, with distinct primes pip_i. Then Ω(n)=e1+e2+⋯+ek\Omega(n) = e_1 + e_2 + \dots + e_k. So Ω(n)\Omega(n) is the number of prime factors of nn, counted with multiplicity.

Remarks

For example, Ω(1)=0\Omega(1) = 0, Ω(12)=3\Omega(12) = 3 and Ω(360)=6\Omega(360) = 6. The function Ω\Omega is completely additive: Ω(mn)=Ω(m)+Ω(n)\Omega(mn) = \Omega(m) + \Omega(n) for all positive mm and nn. The Liouville function is λ(n)=(−1)Ω(n)\lambda(n) = (-1)^{\Omega(n)}.

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Arithmetic function, definition, no evidence levelArithmetic functiondefinitionDivisibility, definition, no evidence levelDivisibilitydefinitionPrime number, definition, no evidence levelPrime numberdefinitionPrime factorization, definition, no evidence levelPrime factorizationdefinitionNumber of prime factors counted with multiplicity, definition, no evidence levelNumber of prime facto…definition
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  1. Arithmetic function definition
  2. Divisibility definition
  3. Prime number definition
  4. Prime factorization definition
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Further reading

  1. Web https://en.wikipedia.org/wiki/Prime_omega_function, lead section
About this page
Tag08420828EY
URIm2://08420828EY
Aliasesbig-omega-function
Kinddefinition
WritingW0 stub
MSC11A25
DefinesNumber of prime factors counted with multiplicity
UsesPrime factorization, Arithmetic function

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