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PrimeOmega[n]

gives the number of prime factors counting multiplicities in n.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Symbolic Manipulation  
Options  
GaussianIntegers  
Applications  
Basic Applications  
Number Theory  
Properties & Relations  
Possible Issues  
Neat Examples  
See Also
Related Guides
History
Cite this Page

PrimeOmega

PrimeOmega[n]

gives the number of prime factors counting multiplicities in n.

Details and Options

  • Integer mathematical function, suitable for both symbolic and numerical manipulation.
  • PrimeOmega gives the number of prime factors of an integer with multiplicity.
  • For a number with a unit and primes, PrimeOmega[n] returns k1+⋯+km.
  • With the setting GaussianIntegers->True, PrimeOmega gives the number of Gaussian prime factors.
  • PrimeOmega[m+In] automatically works over Gaussian integers.

Examples

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Basic Examples  (2)

Compute PrimeOmega at 30:

Plot the PrimeOmega sequence for the first 100 numbers:

Scope  (8)

Numerical Evaluation  (4)

PrimeOmega works over integers:

Gaussian integers:

Compute for large integers:

PrimeOmega threads over lists:

Symbolic Manipulation  (4)

TraditionalForm formatting:

Reduce expressions:

Solve equations:

Identify the PrimeOmega sequence:

Options  (1)

GaussianIntegers  (1)

Compute PrimeOmega over integers:

Gaussian integers:

Applications  (6)

Basic Applications  (2)

Table of the values of PrimeOmega for the integers up to 100:

Histogram of the values of PrimeOmega:

Number Theory  (4)

Use PrimeOmega to test for a prime number:

Use PrimeOmega to compute LiouvilleLambda:

Compare with:

Plot the average over values of PrimeOmega for different ranges of integer arguments:

The Fourier statistics of the PrimeOmega sequence:

Properties & Relations  (5)

Use FactorInteger to find the number of prime factors counting multiplicities:

PrimeOmega is a completely additive function:

PrimeOmega gives the exponent for a prime power:

PrimeOmega and PrimeNu are equivalent when the argument is square-free:

PrimeOmega is always greater than or equal to PrimeNu:

Possible Issues  (1)

PrimeOmega is not defined at 0:

Neat Examples  (2)

Plot the arguments of the Fourier transform of PrimeOmega:

Plot the Ulam Spiral of PrimeOmega:

Wolfram Research (2008), PrimeOmega, Wolfram Language function, https://reference.wolfram.com/language/ref/PrimeOmega.html.

Text

Wolfram Research (2008), PrimeOmega, Wolfram Language function, https://reference.wolfram.com/language/ref/PrimeOmega.html.

CMS

Wolfram Language. 2008. "PrimeOmega." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PrimeOmega.html.

APA

Wolfram Language. (2008). PrimeOmega. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PrimeOmega.html

BibTeX

@misc{reference.wolfram_2025_primeomega, author="Wolfram Research", title="{PrimeOmega}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/PrimeOmega.html}", note=[Accessed: 01-March-2026]}

BibLaTeX

@online{reference.wolfram_2025_primeomega, organization={Wolfram Research}, title={PrimeOmega}, year={2008}, url={https://reference.wolfram.com/language/ref/PrimeOmega.html}, note=[Accessed: 01-March-2026]}