Number of distinct prime factors

Definition

For a positive integer $n$, $\omega(n)$ is the number of distinct prime numbers that divide $n$. So if $n = p_1^{e_1} \cdots p_k^{e_k}$ with distinct primes $p_i$ and $e_i \ge 1$, then $\omega(n) = k$.

Remarks

For example, $\omega(1) = 0$, $\omega(12) = 2$ and $\omega(30) = 3$. The function $\omega$ is additive: $\omega(mn) = \omega(m) + \omega(n)$ for coprime $m$ and $n$. Compare with the number of prime factors with multiplicity $\Omega(n)$. We have $\omega(n) \le \Omega(n)$, with equality exactly for squarefree $n$.

Explanations

L1 human

The function $\omega(n)$ counts the different primes in $n$ and ignores repeats. For $360 = 2^3 \cdot 3^2 \cdot 5$, the different primes are $2$, $3$ and $5$, so $\omega(360) = 3$.