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Auil's conjecture that the iterative sequence enumerates the Möbius numbers

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Let K={ki}i∈N\mathbb K=\{k_i\}_{i\in\mathbb N} be the set of integers produced by the paper's iterative construction, and let

M={m∈N:μ(m)≠0}\mathbb M=\{m\in\mathbb N:\mu(m)\neq0\}

be the set of square-free, or Möbius, numbers. Auil's enumeration conjecture. The iterative construction produces every Möbius number and no others:

K=M.\mathbb K=\mathbb M.

The initial terms displayed in the source agree with the square-free numbers, and numerical evidence is reported in support of the conjecture. Its general validity remains unresolved.

References

Primary source

F. Auil, “A Sequence of Beurling Functions Related to the Natural Approximation B_n Defined by an Iterative Construction Generating Square-Free Numbers k and the Value of the Mobius Function at k”, arXiv:math/0408093 (2004).

mobius-functionnumber-theorysquare-free-numbers
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