# In the definition of the Mobius function and basic properties, can we change square-free to something else?

The Mobius function isolates square-free numbers ie numbers. Can we do this with other numbers with certain properties?

• The question should actually be, if it's useful. Of course you can define other functions. – Paul K Nov 25 '17 at 17:19
• There are also $$k$$-mobius functions, which isolate numbers that don't have a k-th power of a prime in their factorization – Verdruss Nov 25 '17 at 17:58
• You can make whatever definitions you want, but then there's the question of whether they're useful for anything. The Mobius function is useful for all sorts of things, due to its connection to Mobius inversion and the Riemann zeta function; it's singled out by very special considerations that won't apply to more general functions. – Qiaochu Yuan Nov 25 '17 at 18:25