number of distinct prime factors function

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

$$n=\prod _{i=1}^{k}p_i{}^{a_i},$$

where the $p_i$ are distinct primes, there being $k$ of them, and the $a_i$ are positive integers (not necessarily distinct), then $\omega (n)=k$.

Obviously for a prime $p$ it follows that $\omega (p)=1$. When $n$ is a squarefree number, then $\mathrm{\Omega }(n)=\omega (n)$, where $\mathrm{\Omega }(n)$ is the number of (nondistinct) prime factors function (http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction). Otherwise, $\mathrm{\Omega }(n)>\omega (n)$.

$\omega (n)$ is an additive function, and it can be used to define a multiplicative function like the Möbius function $\mu (n)={(-1)}^{\omega (n)}$ (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
Related topic	2omeganLeTaunLe2Omegan