An Algorithm to Generate Square-Free Numbers and to Compute the Moebius Function

TL;DR

This paper presents an algorithm that generates square-free numbers and computes the Moebius function, supported by theoretical proofs and numerical evidence up to 5 million.

Contribution

The paper introduces a novel algorithm that produces all square-free numbers and calculates the Moebius function directly from its outputs, with rigorous proofs of key properties.

Findings

Algorithm produces square-free numbers.

All square-free numbers are generated by the algorithm.

The Moebius function can be computed from the algorithm's outputs.

Abstract

We introduce an algorithm that iteratively produces a sequence of natural numbers k_i and functions b_i. The number k_(i+1) arises as the first point of discontinuity of b_i above k_i. We derive a set of properties of both sequences, suggesting that (1) the algorithm produces square-free numbers k_i, (2) all the square-free numbers are generated as the output of the algorithm, and (3) the value of the Moebius function mu(k_i) can be evaluated as b_i(k_(i+1)) - b_i(k_i). 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 5x10^6, seems to support this conjecture, and shows a total running time linear or quadratic, depending on the implementation.