# 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 |