PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: Very high
number of (nondistinct) prime factors function (Definition)

The 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 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)}$.

The sequence $ \{\Omega(n)\}$ appears in the OEIS as sequence A001222.

The sequence $ \{2^{\Omega(n)}\}$ appears in the OEIS as sequence A061142.



"number of (nondistinct) prime factors function" is owned by Wkbj79.
(view preamble)

View style:

See Also: number of distinct prime factors function, $2^{\omega(n)} \le \tau(n) \le 2^{\Omega(n)}$


Attachments:
$\displaystyle \sum_{n \le x} y^{\Omega(n)}=O\left( \frac{x(\log x)^{y-1}}{2-y} \right)$ for $1 \le y<2$ (Theorem) by Wkbj79
$\displaystyle x\log^2x=O\left(\sum_{n \le x} 2^{\Omega(n)} \right)$ (Theorem) by Wkbj79
Log in to rate this entry.
(view current ratings)

Cross-references: OEIS, sequence, Liouville function, completely multiplicative function, completely additive function, number of distinct prime factors function, number, squarefree, primes, natural number, prime factors
There are 8 references to this entry.

This is version 13 of number of (nondistinct) prime factors function, born on 2006-07-27, modified 2007-04-15.
Object id is 8183, canonical name is NumberOfNondistinctPrimeFactorsFunction.
Accessed 1256 times total.

Classification:
AMS MSC: 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)