# Square-free kernel

From Number

This article defines an arithmetic function or number-theoretic function: a function from the natural numbers to a ring (usually, the ring of integers, rational numbers, real numbers, or complex numbers).

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## Contents

## Definition

Let be a natural number. The **square-free kernel** or **square-free part** of is the product of all the prime divisors of .

## Relation with other arithmetic functions

## Properties

### Multiplicativity

This arithmetic function is a multiplicative function: the product of this function for two natural numbers that are relatively prime is the value of the function at the product.

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### Preservation of divisibility

This arithmetic function is a divisibility-preserving function: if one natural number divides another, the value of the function at the first number also divides the value at the second number.

View other divisibility-preserving functions