SquarePartRoot

The root $s$ of the square part of $n=\pm s^2 q$: the largest $s$ with $s^2$ dividing $n$.

OEISA000188↗

SquarePartRoot(n)$s$ in $|n|=s^2 q$ with $q$ square-free; the largest $s$ whose square divides $n$.

SageMathisqrt(abs(n)//abs(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 $s$, and SquareFreePart is $q$. $n$ is square-free exactly when $s=1$.
• The sign is a unit, as in PrimeOmega: $\mathrm{SquarePartRoot}(-12)=2$. Defined on nonzero integers; 0 stays unevaluated.
• Halves every exponent in the factorization of $|n|$, rounding down.

Plot

Examples

Basic #

$12 = 2^2\cdot3$

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In$\mathrm{SquarePartRoot}(12)$Out$2$

$360 = 6^2\cdot10$

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

A square-free number has no square part: the root is 1

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In$\mathrm{SquarePartRoot}(30)$Out$1$

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

Scope #

Negative $n$: the sign is a unit

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

Listable over a list

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In$\mathrm{SquarePartRoot}(\bigl\lbrack8, 9, 10\bigr\rbrack)$Out$\bigl\lbrack2, 3, 1\bigr\rbrack$

Large integers: $10! = 720^2\cdot7$

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In$\mathrm{SquarePartRoot}(10!)$Out$720$

Applications #

A perfect square's root is its square root, $\sqrt{144}=12$

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In$\mathrm{SquarePartRoot}(144)$Out$12$

Simplifying $\sqrt{1000}=10\sqrt{10}$: the coefficient pulled out is $s$

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

Properties #

$n=s^2 q$. See SquareFreePart

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

$n$ is square-free exactly when $s=1$. See IsSquareFree

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In$\mathrm{SquarePartRoot}(70)=1$Out$\mathrm{True}$

Possible issues #

Zero is outside the domain, so it stays unevaluated

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

About

Examples	12
Checked against	Wolfram Language 11 agree1 differSageMath 11 agree1 error
Crosswalk	OEISSageMath

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 has no head for it; $s=\sqrt{n/q}$ from its squarefree_part, reading |n|.

See also: SquareFreePart, IsSquareFree, FactorInteger, IntegerExponent