Divisors mu Möbius

Möbius inversion formula states that if g and f are arithmetic functions satisfying

g left parenthesis n right parenthesis space equals space sum for d vertical line n of f left parenthesis d right parenthesis

for every integer n greater or equal than 1, then

f left parenthesis n right parenthesis equals sum for d vertical line n of mu left parenthesis d right parenthesis g left parenthesis n over d right parenthesis

for every integer n greater or equal than 1, where mu left parenthesis d right parenthesis is the Möbius function. Phi Euler's function satisfies

n equals sum for d vertical line n of phi left parenthesis d right parenthesis

Then, applying Möbius inversion formula we can write

phi left parenthesis n right parenthesis equals sum for d vertical line n of mu left parenthesis d right parenthesis n over d equals n sum for table row cell d space vertical line space n end cell row cell mu left parenthesis d right parenthesis space not equal to 0 end cell end table of fraction numerator 1 over denominator mu left parenthesis d right parenthesis times d end fraction

divisors_mu_moebius(Integer)

Given an integer n, returns a list with the divisors of n multiplied by mu left parenthesis d right parenthesis.