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,
is square-free. However,
is not square-free: it is divisible by
, and the power of
in the prime factorization is to a power larger than one.
The Möbius function is a function which takes in natural numbers and is usually written as
. The value of
depends on whether or not
is square-free. Specifically,
![{\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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1aa40b808f0da4219564f64145d6b05a7cee2d92)
For example,
is square-free with an 2 prime factors, so
. Since
is not square-free, then
.
Since
is prime, it is its own prime decomposition. That is, the prime factorization of
has 1 prime factor, so
.