The 'Moebious inversion formula' extablishes that if ...

f(n)= Sum[for all d|n] g(d) (1)

... then...

g(n)= Sum[for all d|n] f(d) μ (n/d) (2)

Now if You set in (1) f(n)= σ (n) is...

σ (n)= Sum[for all d|n] d (3)

... so that g(d)=d and (2) becomes...

n= Sum[for all d|n] σ (d) μ (n/d) = Sum[for all d|n] μ (d) σ (n/d) (4)

Kind regards

chi sigma