number of distinct prime factors function

The number of distinct prime factors function ω(n) counts how many distinct prime factors n has. Expressing n as


where the pi are distinct primes, there being k of them, and the ai are positive integers (not necessarily distinct), then ω(n)=k.

Obviously for a prime p it follows that ω(p)=1. When n is a squarefreeMathworldPlanetmath number, then Ω(n)=ω(n), where Ω(n) is the number of (nondistinct) prime factorsMathworldPlanetmath functionMathworldPlanetmath ( Otherwise, Ω(n)>ω(n).

ω(n) is an additive functionMathworldPlanetmath, and it can be used to define a multiplicative functionMathworldPlanetmath like the Möbius function μ(n)=(-1)ω(n) (as long as n is squarefree).

Title number of distinct prime factors function
Canonical name NumberOfDistinctPrimeFactorsFunction
Date of creation 2013-03-22 16:06:52
Last modified on 2013-03-22 16:06:52
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 9
Author CompositeFan (12809)
Entry type Definition
Classification msc 11A25
Related topic NumberOfNondistinctPrimeFactorsFunction
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