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 … 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 this indicator function is
where ζ(s) is the See more

If we represent a square-free number as the infinite product
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 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 less than n are not divisible by 4, 8/9 of these numbers are not divisible by 9, and so on. Because these ratios satisfy the See more

The central binomial coefficient
$${\displaystyle {2n \choose n}}$$
is never squarefree for n > 4. This was proven in 1985 for all … See more