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Number theory
Erdős problem 291 - shiu heuristic density zero
In particular, there should be infinitely many n, but the set of such n should have
density zero. Unfortunately this heuristic is difficult to turn into a proof.
[ErGr80, p.34] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[Sh16] P. Shiu, The denominators of harmonic numbers. arXiv:1607.02863 (2016).
[WuYa22] Wu, Bing-Ling and Yan, Xiao-Hui, On the denominators of harmonic numbers. {IV}. C. R. Math. Acad. Sci. Paris (2022), 53--57.
A statement that does not faithfully capture the original conjecture is the one real risk here, so we would rather hear about it early - before someone spends weeks on it.
Erdős problem 291 - shiu heuristic density zero · Conjectures.io