# 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 $\Omega(n)=\omega(n)$, where $\Omega(n)$ is the number of (nondistinct) prime factors function (http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction). Otherwise, $\Omega(n)>\omega(n)$.

$\omega(n)$ 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 NumberOfDistinctPrimeFactorsFunction 2013-03-22 16:06:52 2013-03-22 16:06:52 CompositeFan (12809) CompositeFan (12809) 9 CompositeFan (12809) Definition msc 11A25 NumberOfNondistinctPrimeFactorsFunction 2omeganLeTaunLe2Omegan