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 /
----
d¦n
Once the inversion has taken place,
----
\
g(n) = > f(d)
/
----
d|n
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__.