SquareFreePart

The square-free part $q$ of $n=\pm s^2 q$: the product of the primes dividing $n$ to an odd power.

WikipediaSquare-free integer↗MathWorldSquarefree↗WikidataQ50706↗OEISA007913↗A005117↗

SquareFreePart(n)$q$ in $|n|=s^2 q$ with $q$ square-free; the sign of $n$ is a unit and is dropped.

SageMathabs(n).squarefree_part()✓↗

Typed signatures

Type	Package	Applies
(integer) -> integer	number-theory	always

Domain: Number theory

Details

• Every nonzero integer is $\pm s^2 q$ for a unique $s\ge1$ and square-free $q\ge1$; this is $q$, and SquarePartRoot is $s$. $n$ is square-free exactly when $s=1$, and a perfect square (up to sign) when $q=1$.
• The sign is a unit, as in PrimeOmega, so $q\ge1$ always: $\mathrm{SquareFreePart}(-12)=3$. Sage's squarefree_part keeps the sign instead.
• Defined on nonzero integers; 0 stays unevaluated.
• Also called the square-free kernel; that name is more often given to the radical, the product of the distinct primes, which is a different function. The radical is Wolfram Function Repository's IntegerRadical.

Plot

Examples

Basic #

$12 = 2^2\cdot3$

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

$18 = 3^2\cdot2$

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In$\mathrm{SquareFreePart}(18)$Out$2$

A square-free number is its own square-free part

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In$\mathrm{SquareFreePart}(30)$Out$30$

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In$\mathrm{SquareFreePart}(1)$Out$1$

Scope #

Negative $n$: the sign is a unit, so the part is positive

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

Listable over a list

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In$\mathrm{SquareFreePart}(\bigl\lbrack8, 9, 10\bigr\rbrack)$Out$\bigl\lbrack2, 1, 10\bigr\rbrack$

Large integers: $10! = 2^8\cdot3^4\cdot5^2\cdot7$

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In$\mathrm{SquareFreePart}(10!)$Out$7$

Applications #

Simplifying $\sqrt{1000}=10\sqrt{10}$: the radicand left over is the square-free part

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In$\mathrm{SquareFreePart}(1\,000)$Out$10$

Properties #

$n=s^2 q$. See SquarePartRoot

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In$360=\mathrm{SquarePartRoot}(360)^2\mathrm{SquareFreePart}(360)$Out$\mathrm{True}$

The part is always square-free. See IsSquareFree

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In$\mathrm{IsSquareFree}(\mathrm{SquareFreePart}(360))$Out$\mathrm{True}$

A perfect square has part 1

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In$\mathrm{SquareFreePart}(144)$Out$1$

Possible issues #

Zero is outside the domain, so it stays unevaluated

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In$\mathrm{SquareFreePart}(0)$Out$\mathrm{SquareFreePart}(0)$

About

Examples	12
Checked against	Wolfram Language 11 agree1 differSageMath 11 agree1 unevaluated
Crosswalk	WikipediaMathWorldWikidataOEIS ×2SageMath

Implementation

mappedwolframpackages/oracle/src/mappings.ts

Wolfram has no head for it; this is its definition over FactorInteger, with the sign of n a unit.

mappedsagepackages/oracle/src/mappings.ts

Sage's squarefree_part keeps the sign of n; this reads |n|.

See also: SquarePartRoot, IsSquareFree, FactorInteger, MoebiusMu