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'number of (nondistinct) prime factors function'
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| Title of object: |
number of (nondistinct) prime factors function |
| Canonical Name: |
NumberOfNondistinctPrimeFactorsFunction |
| Type: |
Definition |
| Created on: |
2006-07-27 19:23:14 |
| Modified on: |
2006-07-28 22:23:33 |
| Classification: |
msc:11A25 |
Preamble:
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Content:
The \PMlinkescapetext{{\em number of (nondistinct) prime factors function}} $\Omega(n)$ 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 an additive function and thus can be exponentiated to define a multiplicative function. For example, the Liouville function can be defined as $\lambda(n) = (-1)^{\Omega(n)}$. |
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