| Version current |
Version 12 |
| 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 $\displaystyle \Omega(n)=\sum_{j=1}^k a_j$. |
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 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 $\lambda(n) = (-1)^{\Omega(n)}$.
|
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 $\lambda(n) = (-1)^{\Omega(n)}$.
|
|
|
| The sequence $\{\Omega(n)\}$ appears in the OEIS as sequence \PMlinkexternal{A001222}{http://www.research.att.com/~njas/sequences/?q=A001222}. |
The sequence $\{\Omega(n)\}$ appears in the OEIS as sequence \PMlinkexternal{A001222}{http://www.research.att.com/~njas/sequences/?q=A001222}. |
|
|
| The sequence $\{2^{\Omega(n)}\}$ appears in the \PMlinkname{OEIS}{OEIS} as sequence \PMlinkexternal{A061142}{http://www.research.att.com/~njas/sequences/?q=A061142}. |
The sequence $\{2^{\Omega(n)}\}$ appears in the \PMlinkname{OEIS}{OEIS} as sequence \PMlinkexternal{A061142}{http://www.research.att.com/~njas/sequences/?q=A061142}. |
