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number of distinct prime factors function (Definition)

The number of distinct prime factors function $ \omega(n)$ counts how many distinct prime factors $ n$ has. Expressing $ n$ as

$\displaystyle n = \prod_{i = 1}^k {p_i}^{a_i},$
where the $ p_i$ are distinct primes, there being $ k$ of them, and the $ a_i$ are positive integers (not necessarily distinct), then $ \omega(n) = k$.

Obviously for a prime $ p$ it follows that $ \omega(p) = 1$. When $ n$ is a squarefree number, then $ \Omega(n) = \omega(n)$, where $ \Omega(n)$ is the number of (nondistinct) prime factors function. Otherwise, $ \Omega(n) > \omega(n)$.

$ \omega(n)$ is an additive function, and it can be used to define a multiplicative function like the Möbius function $ \mu(n) = (-1)^{\omega(n)}$ (as long as $ n$ is squarefree).



"number of distinct prime factors function" is owned by CompositeFan.
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See Also: number of (nondistinct) prime factors function, $2^{\omega(n)} \le \tau(n) \le 2^{\Omega(n)}$


Attachments:
$\displaystyle \sum_{n \le x} y^{\omega(n)}=O_y(x(\log x)^{y-1})$ for $y \ge 0$ (Theorem) by Wkbj79
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Cross-references: Möbius function, multiplicative function, additive function, number, squarefree, integers, positive, primes, prime factors
There are 15 references to this entry.

This is version 6 of number of distinct prime factors function, born on 2006-07-27, modified 2006-07-28.
Object id is 8180, canonical name is NumberOfDistinctPrimeFactorsFunction.
Accessed 2194 times total.

Classification:
AMS MSC: 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)
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