Erdős problem 316 · problems.science

Is it true that if $A \subseteq \mathbb{N}\setminus\{1\}$ is a finite set with $\sum_{n \in A} \frac{1}{n} < 2$ then there is a partition $A=A_1 \sqcup A_2$ such that $\sum_{n \in A_i} \frac{1}{n} < 1$ for $i=1,2$ ?

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FormalConjectures/ErdosProblems/

316.lean

Retained formal statement2 of 3

This is not true in general, as shown by Sándor [Sa97], who observed that the proper divisors of $120$ form a counterexample. More generally, Sándor shows that for any $n\geq 2$ there exists a finite set $A\subseteq \mathbb{N}\backslash\{1\}$ with $\sum_{k\in A}\frac{1}{k} < n$ and no partition into $n$ parts each of which has $\sum_{k\in A_i}\frac{1}{k}<1$ .

FormalConjectures/ErdosProblems/316.leanErdos316.erdos_316.variants.generalized6 linesExact file

1∀ (n : ℕ),2  2 ≤ n →3    ∃ A,4      A.Nonempty ∧5        0 ∉ A ∧6          1 ∉ A ∧ ∑ k ∈ A, 1 / ↑k < ↑n ∧ ∀ (P : Finpartition A), P.parts.card = n → ∃ p ∈ P.parts, 1 ≤ ∑ n ∈ p, 1 / ↑n

SolvedStatement only, no proof

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