Square-free integer

Definition

A nonzero integer $n$ is square-free when no square of a prime number divides $n$. That is, there is no prime $p$ such that $p^2 \mid n$.

Remarks

Equivalently, $n$ is square-free when the only perfect squares that divide $n$ are $1$. A positive integer is square-free if and only if each exponent in its prime factorization is $1$. The first square-free positive integers are $1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15$. The integer $12 = 2^2 \cdot 3$ is not square-free. The integer $0$ is not square-free, because each square divides $0$.

Explanations

L1 human

A number is square-free when no prime appears twice in its factorization. For example, $30 = 2 \cdot 3 \cdot 5$ is square-free, but $18 = 2 \cdot 3 \cdot 3$ is not, because $9$ divides $18$.