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Homenumber of distinct prime factors function

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# 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. 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).

Related:

NumberOfNondistinctPrimeFactorsFunction, 2omeganLeTaunLe2Omegan

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