Square-free integer
A square-free integer is a number which is not divisible by any square numbers other than 1. In other words, each prime number that appears in its prime factorization appears exactly once.
For example, [math]\displaystyle{ 6 = 2\times3 }[/math] is square-free. However, [math]\displaystyle{ 27 = 3^3 }[/math] is not square-free: it is divisible by [math]\displaystyle{ 9 = 3^2 }[/math], and the power of [math]\displaystyle{ 3 }[/math] in the prime factorization is to a power larger than one.
Möbius function
The Möbius function is a function which takes in natural numbers and is usually written as [math]\displaystyle{ \mu(n) }[/math]. The value of [math]\displaystyle{ \mu(n) }[/math] depends on whether or not [math]\displaystyle{ n }[/math] is square-free. Specifically,
- [math]\displaystyle{ \mu(n) = \begin{cases} 1, &\text{if }n\text{ is square-free and has an even number of prime factors}, \\ -1, &\text{if }n\text{ is square-free and has an odd number of prime factors} \\ 0, &\text{if }n\text{ is not square-free} \end{cases} }[/math]
For example, [math]\displaystyle{ 6 = 2\times 3 }[/math] is square-free with an 2 prime factors, so [math]\displaystyle{ \mu(6) = 1 }[/math]. Since [math]\displaystyle{ 27 = 3^3 }[/math] is not square-free, then [math]\displaystyle{ \mu(27) = 0 }[/math].
Since [math]\displaystyle{ 5 }[/math] is prime, it is its own prime decomposition. That is, the prime factorization of [math]\displaystyle{ 5 }[/math] has 1 prime factor, so [math]\displaystyle{ \mu(5) = -1 }[/math].
Square-free Integer Media
10 is square-free, as its divisors greater than 1 are 2, 5, and 10, none of which is square (the first few squares being 1, 4, 9, and 16)
Square-free integers up to 120 remain after eliminating multiples of squares of primes up to √120
