Computing the (number of) inverses of Euler's totient and other multiplicative functions.

We propose a generic algorithm for computing the inverses of a multiplicative function. We illustrate our algorithm with Euler's totient function and the sum of k-th powers of divisors. Our approach can be further adapted for computing certain functions of the inverses, such as their quantity, the smallest/largest inverse, which may be computed without and possibly faster than the inverses themselves.