PrimeNu

The number of distinct prime factors of n.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaPrime omega function↗MathWorldDistinctPrimeFactors↗WikidataQ55630466↗DLMF27.2.i.p1↗OEISA001221↗Wolfram LanguagePrimeNu✓↗

PrimeNu(n)number of distinct prime factors of $n$.

SageMathlen(prime_divisors(n))SymPyprimenu(n)Juliaenumeratio_prime_nu(n)Rustprime_nu(n)

Typed signatures

Type	Package	Applies
(integer) -> integer	compute-engine	always
(number | quadratic_integer, any*) -> integer	number-theoryreplaces compute-engine	always

Domain: Number theory

Details

• Counts distinct primes only -- $\nu(2^5)=1$, not 5.
• Equal to the length of FactorInteger$(n)$.
• $\nu(1)=0$: 1 has no prime factors.
• Always $\nu(n)\le\Omega(n)$, with equality exactly when $n$ is squarefree. See PrimeOmega.

Examples

Basic #

live ↗

In$\mathrm{PrimeNu}(24)$Out$2$

live ↗

In$\mathrm{PrimeNu}(360)$Out$3$

live ↗

In$\mathrm{PrimeNu}(105)$Out$3$

Scope #

Threads element-wise over a list, as Wolfram's Listable heads do

live ↗

In$\mathrm{PrimeNu}(\bigl\lbrack4, 28, 180\bigr\rbrack)$Out$\bigl\lbrack1, 2, 3\bigr\rbrack$

Large integers: the 15 primes up to 50 divide $50!$

live ↗

In$\mathrm{PrimeNu}(50!)$Out$15$

Gaussian integers: $3+i=(1+i)(2-i)$

live ↗

In$\mathrm{PrimeNu}(3+i)$Out$2$

GaussianIntegers -> True: $5$ splits, so $105$ has 4 Gaussian prime factors

live ↗

In$\mathrm{PrimeNu}(105, \mathrm{Rule}(\mathrm{Over}, \mathrm{GaussianIntegers}))$Out$4$

Listable over a Range

live ↗

In$\mathrm{PrimeNu}(1..20)$Out$\bigl\lbrack0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2\bigr\rbrack$

Counts distinct primes: $2,3,5$

live ↗

In$\mathrm{PrimeNu}(30)$Out$3$

Negative $n$: the sign is a unit, so $\omega(-12)=\omega(12)$

live ↗

In$\mathrm{PrimeNu}(-12)$Out$2$

Applications #

For squarefree $n$, $\mu(n)=(-1)^{\omega(n)}$

live ↗

In$\mathrm{MoebiusMu}(42)=(-1)^{\mathrm{PrimeNu}(42)}$Out$\mathrm{True}$

Squarefree numbers also have $\lambda(n)=(-1)^{\omega(n)}$

live ↗

In$\mathrm{LiouvilleLambda}(42)=(-1)^{\mathrm{PrimeNu}(42)}$Out$\mathrm{True}$

Properties #

PrimeNu(n) is the length of FactorInteger(n)

live ↗

In$\mathrm{PrimeNu}(50)=\mathrm{Length}(\mathrm{FactorInteger}(50))$Out$\mathrm{True}$

$50=2\cdot5^2$: repeated primes count once

live ↗

In$\mathrm{PrimeNu}(50)$Out$2$

Additive over coprime factors $9$ and $40$

live ↗

In$\mathrm{PrimeNu}(360)=\mathrm{PrimeNu}(9)+\mathrm{PrimeNu}(40)$Out$\mathrm{True}$

A prime square has one distinct prime

live ↗

In$\mathrm{PrimeNu}(49)$Out$1$

Equal to PrimeOmega when $n$ is squarefree

live ↗

In$\mathrm{PrimeNu}(210)=\mathrm{PrimeOmega}(210)$Out$\mathrm{True}$

Squarefree $33=3\cdot11$ gives $\mu=1$, $\omega=2$

live ↗

In$\mathrm{MoebiusMu}(33)=(-1)^{\mathrm{PrimeNu}(33)}$Out$\mathrm{True}$

Same parity rule for $\lambda$ at a squarefree number

live ↗

In$\mathrm{LiouvilleLambda}(33)=(-1)^{\mathrm{PrimeNu}(33)}$Out$\mathrm{True}$

A prime has $\omega=1$

live ↗

In$\mathrm{PrimeNu}(2)$Out$1$

Known value · Wolfram MathematicalFunctionData, PrimeNu

Again one distinct prime

live ↗

In$\mathrm{PrimeNu}(3)$Out$1$

Known value · Wolfram MathematicalFunctionData, PrimeNu

$6=2\cdot3$

live ↗

In$\mathrm{PrimeNu}(6)$Out$2$

Known value · Wolfram MathematicalFunctionData, PrimeNu

$2^2$ has just one distinct prime

live ↗

In$\mathrm{PrimeNu}(4)$Out$1$

Known value · Wolfram MathematicalFunctionData, PrimeNu

$\omega(n)\le\Omega(n)$ for every $n\le99$

live ↗

In$\mathrm{All}(1..100, \operatorname{\_1}↦\mathrm{PrimeNu}(\operatorname{\_1})≤\mathrm{PrimeOmega}(\operatorname{\_1}))$Out$\mathrm{True}$

Possible issues #

$32=2^5$ has just one distinct prime factor despite the high power

live ↗

In$\mathrm{PrimeNu}(32)$Out$1$

1 has no prime factors

live ↗

In$\mathrm{PrimeNu}(1)$Out$0$

About

Examples	26
Checked against	Wolfram Language 26 agreeSageMath 23 agreeSymPy 19 agree2 errormathlib4 20 agree3 errorJulia 21 agree1 errorRust 15 agree2 error
Crosswalk	WikipediaMathWorldWikidataDLMFOEISWolfram LanguageSageMathSymPyJuliaRust

Implementation

mappedsympypackages/oracle/src/mappings.ts

primenu counts distinct prime factors of an integer; a Gaussian integer is not one.

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

The number of primes in the factorization.

mappedmathlib4packages/oracle/src/mappings.ts

ArithmeticFunction.cardDistinctFactors, omega(n): distinct prime factors, omega(0) = 0; natural n only.

See also: PrimeOmega, FactorInteger