Let g(n) be a function with domain and range in the natural numbers. The Möbius inversion formula is as follows:

       \        / n \
f(n) =  >  u(d)g| - |
       /        \ d /

Once the inversion has taken place,

g(n) =  >   f(d)

also holds.

In other words, the inversion formula is the sum over the divisors of n of g(n/d); every inversion may be 'undone' as well. The use of these inversions leads into cyclotomic polynomials and some factoring tricks there.

All information discovered at http://mathworld.wolfram.com/MoebiusInversionFormula.html.