The radical of an integer is its largest square-free factor, that is $${\displaystyle \textstyle \prod _{i=1}^{k}q_{i}}$$ with notation of the preceding section. An integer is square-free if and only if it is equal to its radical. See more

The absolute value of the Möbius function is the indicator function for the square-free integers – that is, |μ(n)| is equal to 1 if n is square-free, and 0 if it is not. The Dirichlet series of … See more

If we represent a square-free number as the infinite product
$${\displaystyle \prod _{n=0}^{\infty }(p_{n+1})^{a_{n}},a_{n}\in \lbrace 0,1\rbrace ,{\text{ and }}p_{n}{\text{ is the }}n{\text{th prime}},}$$
then we may take … See more

Let us call "t-free" a positive integer that has no t-th power in its divisors. In particular, the 2-free integers are the square-free integers.
The multiplicative function $${\displaystyle \mathrm {core} _{t}(n)}$$ maps every positive integer n to … See more

A positive integer $${\displaystyle n}$$ is square-free if and only if in the prime factorization of $${\displaystyle n}$$, no prime factor occurs … See more

Let Q(x) denote the number of square-free integers between 1 and x (OEIS: A013928 shifting index by 1). For large n, 3/4 of the positive integers … See more

The central binomial coefficient
$${\displaystyle {2n \choose n}}$$
is never squarefree for n > 4. This was proven in 1985 for all sufficiently large integers by András Sárközy, and for all integers > 4 in 1996 by Olivier Ramaré and See more