# number of (nondistinct) prime factors function

The $\mathrm{\Omega}(n)$ counts with repetition how many prime factors^{} a natural number^{} $n$ has. If $n={\displaystyle \prod _{j=1}^{k}}p_{j}{}^{{a}_{j}}$ where the $k$ primes ${p}_{j}$ are distinct and the ${a}_{j}$ are natural numbers, then $\mathrm{\Omega}(n)={\displaystyle \sum _{j=1}^{k}}{a}_{j}$.

Note that, if $n$ is a squarefree^{} number, then $\omega (n)=\mathrm{\Omega}(n)$, where $\omega (n)$ is the number of distinct prime factors function. Otherwise, $$.

Note also that $\mathrm{\Omega}(n)$ is a completely additive function and thus can be exponentiated to define a completely multiplicative function^{}. For example, the Liouville function^{} can be defined as $\lambda (n)={(-1)}^{\mathrm{\Omega}(n)}$.

The sequence^{} $\{\mathrm{\Omega}(n)\}$ appears in the OEIS as sequence http://www.research.att.com/ njas/sequences/?q=A001222A001222.

The sequence $\{{2}^{\mathrm{\Omega}(n)}\}$ appears in the OEIS (http://planetmath.org/OEIS) as sequence http://www.research.att.com/ njas/sequences/?q=A061142A061142.

Title | number of (nondistinct) prime factors function^{} |
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Canonical name | NumberOfnondistinctPrimeFactorsFunction |

Date of creation | 2013-03-22 16:07:00 |

Last modified on | 2013-03-22 16:07:00 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 16 |

Author | Wkbj79 (1863) |

Entry type | Definition |

Classification | msc 11A25 |

Related topic | NumberOfDistinctPrimeFactorsFunction |

Related topic | 2omeganLeTaunLe2Omegan |