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
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