Math 2.0

Number theory Definition

Number of distinct prime factors

Builds on Prime numberArithmetic function

Definition

For a positive integer nn, ω(n)\omega(n) is the number of distinct prime numbers that divide nn. So if n=p1e1⋯pkekn = p_1^{e_1} \cdots p_k^{e_k} with distinct primes pip_i and ei≥1e_i \ge 1, then ω(n)=k\omega(n) = k.

Remarks

For example, ω(1)=0\omega(1) = 0, ω(12)=2\omega(12) = 2 and ω(30)=3\omega(30) = 3. The function ω\omega is additive: ω(mn)=ω(m)+ω(n)\omega(mn) = \omega(m) + \omega(n) for coprime mm and nn. Compare with the number of prime factors with multiplicity Ω(n)\Omega(n). We have ω(n)≤Ω(n)\omega(n) \le \Omega(n), with equality exactly for squarefree nn.

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Arithmetic function, definition, no evidence levelArithmetic functiondefinitionDivisibility, definition, no evidence levelDivisibilitydefinitionPrime number, definition, no evidence levelPrime numberdefinitionNumber of distinct prime factors, definition, no evidence levelNumber of distinct pr…definition
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  1. Arithmetic function definition
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Further reading

  1. Web https://en.wikipedia.org/wiki/Prime_omega_function, lead section
About this page
Tag0YVVXT3A0B
URIm2://0YVVXT3A0B
Aliaseslittle-omega-function
Kinddefinition
WritingW0 stub
MSC11A25
DefinesNumber of distinct prime factors
UsesPrime number, Arithmetic function

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