An Algorithm to Generate Square-Free Numbers and to Compute the Moebius Function
Abstract
We introduce an algorithm that iteratively produces a sequence of natural numbers ki and functions bi. The number k(i+1) arises as the first point of discontinuity of bi above ki. We derive a set of properties of both sequences, suggesting that (1) the algorithm produces square-free numbers ki, (2) all the square-free numbers are generated as the output of the algorithm, and (3) the value of the Moebius function mu(ki) can be evaluated as bi(k(i+1)) - bi(ki). The logical equivalence of these properties is rigorously proved. The question remains open if one of these properties can be derived from the definition of the algorithm. Numerical evidence, limited to 5x106, seems to support this conjecture, and shows a total running time linear or quadratic, depending on the implementation.