What is the size of the largest $A \subseteq {1, \dots, N}$ such that all sums $\sum_{n \in S} \frac{1}{n}$ are distinct for $S \subseteq A$ ?

Open — best to date is a honest null, not yet sealed.

number theory · open · formalized (Lean) · 1 attempt

use this record

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

honest null

needs verification

attempted via frontier 'sidon/B2' (transfer_strength=weak) -> no_progress

No solve/partial on this pass. Transfer into the owned frontier was 'weak'. Do not re-attack cold; needs a new idea or richer accumulated context.

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let $R(N)$ denote the maximum size of a set (A\subseteq{1,\dots,N}) such that the map [ S\subseteq A \ \longmapsto\ \sum_{n\in S}\frac1n ] is injective (i.e., all these subset–reciprocal sums are distinct).

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

candidate solution ↗

formal

AMS 11 · open (literature)

theorem erdos_321 (N : ℕ) :
    R N = answer(sorry)

formal-conjectures/321.lean ↗

oeis

A384927 — a(n) is the maximum size of a subset S of {1, 2, ..., n} such that for any distinct elements t, u in S, t + u does not divide t*u.1,2,3,4,5,5,6,7,8,9,10,10,11,12,12,13,14,14,15,15,16,17,18,18,19,20,21,21,22,23,24,25,26,27,27,28,29,30,31,31,32,32,33,3 A391592 — Maximum size of a subset S of {1..n} such that all subset sums of {1/k : k in S} are distinct.1,2,3,4,5,5,6,7,8,9,10,10,11,12,12,13,14,14,15,15,15,16,17,17,18,19,20,20,21,21,22,23,23,24,24,25,26,27,28,28,29,29,30,3

links

#327Suppose

A \subseteq {1, \dots, N}

is such that if

a, b \in A

and

a \neq = b

then

a + b ∤ ab

. Can

A

be 'substantially more' than the odd numbers?What if

a, b \in A

with

a \neq = b

implies

a + b ∤ 2 ab

? Must

∣ A ∣ = o (N)

?A384927

status