PrimeNu [n]
gives the number of distinct primes in n.
PrimeNu
PrimeNu [n]
gives the number of distinct primes in n.
Details and Options
- Integer mathematical function, suitable for both symbolic and numerical manipulation.
- PrimeNu gives the number of distinct prime factors.
- For a number with a unit and primes, PrimeNu [n] returns m.
- With the setting GaussianIntegers->True , PrimeNu gives the number of Gaussian prime factors.
- PrimeNu [m+I n] automatically works over Gaussian integers.
Examples
open all close allScope (9)
Numerical Evaluation (4)
Symbolic Manipulation (5)
TraditionalForm formatting:
Reduce expressions:
Solve equalities:
Identify the PrimeNu sequence:
Dirichlet generating function of 2^PrimeNu :
Compare with DirichletTransform :
Options (1)
GaussianIntegers (1)
Compute PrimeNu over integers:
Gaussian integers:
Applications (7)
Basic Applications (2)
Number Theory (5)
Use PrimeNu to test for a prime power:
Use PrimeNu to compute MoebiusMu and LiouvilleLambda for square-free numbers:
PrimeNu is related to MoebiusMu through the following formula:
Plot the average over values of PrimeNu for different ranges of integer arguments:
The Fourier statistics of the PrimeNu sequence:
Properties & Relations (6)
Use FactorInteger to find the number of distinct prime factors:
PrimeNu is an additive function:
PrimeNu gives 1 for a prime power:
PrimeNu and PrimeOmega are equivalent when the argument is square free:
PrimeNu is always smaller than or equal to PrimeOmega :
If n is square free, PrimeNu is related to MoebiusMu and LiouvilleLambda :
Possible Issues (1)
PrimeNu is not defined at 0:
Related Guides
History
Text
Wolfram Research (2008), PrimeNu, Wolfram Language function, https://reference.wolfram.com/language/ref/PrimeNu.html.
CMS
Wolfram Language. 2008. "PrimeNu." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PrimeNu.html.
APA
Wolfram Language. (2008). PrimeNu. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PrimeNu.html
BibTeX
@misc{reference.wolfram_2025_primenu, author="Wolfram Research", title="{PrimeNu}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/PrimeNu.html}", note=[Accessed: 04-December-2025]}
BibLaTeX
@online{reference.wolfram_2025_primenu, organization={Wolfram Research}, title={PrimeNu}, year={2008}, url={https://reference.wolfram.com/language/ref/PrimeNu.html}, note=[Accessed: 04-December-2025]}