PrimeOmega

The number of prime factors of n, counted with multiplicity.

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PrimeOmega(n)number of prime factors of $n$, counted with multiplicity.

SageMathsum(e for _, e in factor(n))SymPyprimeomega(n)Juliaenumeratio_prime_omega(n)Rustprime_omega(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 every prime factor with multiplicity -- $\Omega(2^5)=5$, unlike PrimeNu's 1.
• Completely additive: $\Omega(mn)=\Omega(m)+\Omega(n)$ for all $m,n$, not just coprime ones.
• $\Omega(1)=0$: 1 has no prime factors.
• $\Omega(n)=\nu(n)$ exactly when $n$ is squarefree; otherwise $\Omega(n)>\nu(n)$.

Examples

Basic #

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

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In$\mathrm{PrimeOmega}(360)$Out$6$

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In$\mathrm{PrimeOmega}(12)$Out$3$

Scope #

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

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In$\mathrm{PrimeOmega}(\bigl\lbrack4, 12, 24\bigr\rbrack)$Out$\bigl\lbrack2, 3, 4\bigr\rbrack$

Large integers

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In$\mathrm{PrimeOmega}(30!)$Out$59$

Gaussian integers: $5+9i$ has norm $106=2\cdot53$, so two prime factors

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In$\mathrm{PrimeOmega}(5+9i)$Out$2$

GaussianIntegers -> True: $12=-(1+i)^4\cdot3$

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In$\mathrm{PrimeOmega}(12, \mathrm{Rule}(\mathrm{Over}, \mathrm{GaussianIntegers}))$Out$5$

Listable over a Range

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In$\mathrm{PrimeOmega}(1..20)$Out$\bigl\lbrack0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3\bigr\rbrack$

$28=2^2\cdot7$ counts $2,2,7$

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In$\mathrm{PrimeOmega}(28)$Out$3$

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

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In$\mathrm{PrimeOmega}(-12)$Out$3$

Properties #

Completely additive: $\Omega(mn)=\Omega(m)+\Omega(n)$ for all $m,n$

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In$\mathrm{PrimeOmega}(24×40)=\mathrm{PrimeOmega}(24)+\mathrm{PrimeOmega}(40)$Out$\mathrm{True}$

$\Omega(n)\ge\nu(n)$ always; here $8=2^3$ gives $\Omega=3$ but $\nu=1$. See PrimeNu

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In$\mathrm{PrimeOmega}(8)≥\mathrm{PrimeNu}(8)$Out$\mathrm{True}$

$100=2^2\cdot5^2$: repeated primes count each time

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In$\mathrm{PrimeOmega}(100)$Out$4$

$16807=7^5$, five factors from one prime

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In$\mathrm{PrimeOmega}(16\,807)$Out$5$

Squarefree $165=3\cdot5\cdot11$, so it matches PrimeNu

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In$\mathrm{PrimeOmega}(165)=\mathrm{PrimeNu}(165)$Out$\mathrm{True}$

Never smaller than the distinct-prime count

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

Possible issues #

1 has no prime factors

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

About

Examples	17
Checked against	Wolfram Language 17 agreeSageMath 13 agreeSymPy 12 agree2 errormathlib4 10 agree3 errorJulia 12 agree1 errorRust 11 agree2 error
Crosswalk	WikipediaMathWorldWikidataOEISWolfram LanguageSageMathSymPyJuliaRust

Implementation

mappedsympypackages/oracle/src/mappings.ts

primeomega counts prime factors with multiplicity 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, with multiplicity.

mappedmathlib4packages/oracle/src/mappings.ts

ArithmeticFunction.cardFactors, Omega(n): prime factors with multiplicity, Omega(0) = 0; natural n only.

See also: PrimeNu, FactorInteger