Kind of a closed form for the Möbius function as the number of points on n-dimensional hyperboloids.

Question

Through experimenting with the Mertens function sum I wrote in Wikipedia I found this formula if you like, for the Möbius function as a sum of the number of points on n-dimensional hyperboloids:

$$\mu(1)=1$$ $$\underset{n > 1}{\mu(n)} = - \underset{a = n}{\sum_{a \geq 2}} 1 + \underset{ab = n}{\sum_{a \geq 2} \sum_{b \geq 2}} 1 - \underset{abc = n}{\sum_{a \geq 2} \sum_{b \geq 2} \sum_{c \geq 2}} 1 + \underset{abcd = n}{\sum_{a \geq 2} \sum_{b \geq 2} \sum_{c \geq 2} \sum_{d \geq 2}} 1 - \cdots$$

$$=1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, 1, -1, 0, 0,..$$

Associated Mathematica program:

nn = 25;
Table[-Sum[If[a == n, 1, 0], {a, 2, nn}] + 
  Sum[Sum[If[a*b == n, 1, 0], {a, 2, nn}], {b, 2, nn}] - 
  Sum[Sum[Sum[If[a*b*c == n, 1, 0], {a, 2, nn}], {b, 2, nn}], {c, 2, 
    nn}] + Sum[
   Sum[Sum[Sum[If[a*b*c*d == n, 1, 0], {a, 2, nn}], {b, 2, nn}], {c, 
     2, nn}], {d, 2, nn}], {n, 1, nn}]
MoebiusMu[Range[25]]
%% - %

How does one prove this formula for $\mu(n)$?