Definition:Radical of Integer

From ProofWiki
Jump to navigation Jump to search

Definition

The radical of an integer $n \in \Z$ is the product of the individual prime factors of $n$.


The radicals of the first few integers are given here:


$n$ Decomposition $\map {\operatorname {rad} } n$
$1$ $1$ $1$
$2$ $2$ $2$
$3$ $3$ $3$
$4$ $2^2$ $2$
$5$ $5$ $5$
$6$ $2 \times 3$ $6$
$7$ $7$ $7$
$8$ $2^3$ $2$
$9$ $3^2$ $3$
$10$ $2 \times 5$ $10$
$11$ $11$ $11$
$12$ $2^2 \times 3$ $6$
$13$ $13$ $13$
$14$ $2 \times 7$ $14$
$15$ $3 \times 5$ $15$
$16$ $2^4$ $2$

This sequence is A007947 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


The radical of $n$ can alternatively be described as the largest square-free integer which divides $n$.


Also known as

The radical of an integer is also known as the square-free kernel.


Also see


Generalizations