The radical of an integer is its largest square-free factor, that is with notation of the preceding section. An integer is square-free
Every … See more

A positive integer is square-free if and only if in the prime factorization of , no prime factor occurs with an exponent larger than one. Another way of stating the same is that for every prime factor of , the prime does not evenly divide . … 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) i… 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 … See more