number of (nondistinct) prime factors function

The Ω(n) counts with repetition how many prime factorsMathworldPlanetmath a natural numberMathworldPlanetmath n has. If n=j=1kpjaj where the k primes pj are distinct and the aj are natural numbers, then Ω(n)=j=1kaj.

Note that, if n is a squarefreeMathworldPlanetmath number, then ω(n)=Ω(n), where ω(n) is the number of distinct prime factors function. Otherwise, ω(n)<Ω(n).

Note also that Ω(n) is a completely additive function and thus can be exponentiated to define a completely multiplicative functionMathworldPlanetmath. For example, the Liouville functionDlmfMathworldPlanetmath can be defined as λ(n)=(-1)Ω(n).

The sequenceMathworldPlanetmath {Ω(n)} appears in the OEIS as sequence njas/sequences/?q=A001222A001222.

The sequence {2Ω(n)} appears in the OEIS ( as sequence njas/sequences/?q=A061142A061142.

Title number of (nondistinct) prime factors functionMathworldPlanetmath
Canonical name NumberOfnondistinctPrimeFactorsFunction
Date of creation 2013-03-22 16:07:00
Last modified on 2013-03-22 16:07:00
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 16
Author Wkbj79 (1863)
Entry type Definition
Classification msc 11A25
Related topic NumberOfDistinctPrimeFactorsFunction
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