---
tag: 08420828EY
kind: definition
name: Number of prime factors counted with multiplicity
aliases: [big-omega-function]
msc: [11A25]
defines: 0KXQKES075
uses: [0AEKDFCYFA, 0H2EPRZC0D]
sources:
  - url: https://en.wikipedia.org/wiki/Prime_omega_function
    locator: lead section
---

## Definition

Let $n$ be a positive integer with [[prime factorization|0AEKDFCYFA]] $n = p_1^{e_1} \cdots p_k^{e_k}$, with distinct primes $p_i$.
Then $\Omega(n) = e_1 + e_2 + \dots + e_k$.
So $\Omega(n)$ is the number of prime factors of $n$, counted with multiplicity.

## Remarks

For example, $\Omega(1) = 0$, $\Omega(12) = 3$ and $\Omega(360) = 6$.
The function $\Omega$ is completely additive: $\Omega(mn) = \Omega(m) + \Omega(n)$ for all positive $m$ and $n$.
The [[Liouville function|08RQAPXF1T]] is $\lambda(n) = (-1)^{\Omega(n)}$.

## Explanations

### L1 human

The function $\Omega(n)$ counts all the prime factors of $n$, with repeats.
For $12 = 2 \cdot 2 \cdot 3$, there are three factors, so $\Omega(12) = 3$, while $\omega(12) = 2$.