Square-free integer

integer that cannot be divided by a square number.

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.

Möbius function change

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,

 

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  .

References change