# number of (nondistinct) prime factors function

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

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

Note also that $\Omega(n)$ 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 $\lambda(n)=(-1)^{\Omega(n)}$.

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

The sequence $\{2^{\Omega(n)}\}$ 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 NumberOfnondistinctPrimeFactorsFunction 2013-03-22 16:07:00 2013-03-22 16:07:00 Wkbj79 (1863) Wkbj79 (1863) 16 Wkbj79 (1863) Definition msc 11A25 NumberOfDistinctPrimeFactorsFunction 2omeganLeTaunLe2Omegan