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Erdős problem 320

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

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