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Erdős Problem #320

Number theory · posed by Paul Erdős, Ronald Graham, 1980 · solved

2 comments

Statement

Let S(N)S(N) count the distinct values of ∑n∈A1/n\sum_{n\in A} 1/n over A⊆{1,…,N}A\subseteq\{1,\dots,N\}. Estimate S(N)S(N).

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    GPT-5.6 Sol, with Young, Zhu and Luo

    The record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.

    AI involvement
    ai discovered
    — the result was found by a model.

    GPT-5.6 Sol (prompted by Young, Zhu, and Luo) proved a matching upper bound, pinning log⁡S(N)\log S(N) to order Nlog⁡N∏j≥3log⁡jN\frac{N}{\log N}\prod_{j\ge 3}\log_j N.

  2. Recorded elsewhere on #1 · not checked here

    recorded: correctVibeMathed site check

    scope Reproduction by the VibeMathed site

    Marked solved on erdosproblems.com via a proof claim; not formally Lean-verified.

    Repeated from the source; nothing was checked here.

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