Erdős Problem #320

Precise statement

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

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

GPT-5.6 Sol

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

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

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

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-07-01

   erdosproblems.com

   VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.