Let $1 \leq a_{1} < \dots < a_{k} \leq n$ be integers such that all sums of the shape $\sum_{u \leq i \leq v} a_{i}$ are distinct. Let $f (n)$ be the maximal such $k$ .How does $f (n)$ grow? Is $f (n) = o (n)$ ?

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

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

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

honest null

needs verification

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

No solve/partial on this pass. Transfer into the owned frontier was 'partial'. 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

Write the prefix sums [ s_0:=0,\qquad s_j:=a_1+\cdots+a_j\quad (1\le j\le k). ] Then every “consecutive sum” [ a_u+\cdots+a_v = s_v-s_{u-1} ] is a **difference of two prefix sums**. So your condition (“all consecutive sums are distinct”) is exactly the statement that all differences [ s_i-s_j\quad (0\le j<i\le k) ] are…

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_357.parts.i : (fun n ↦ (f n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ))

formal-conjectures/357.lean ↗

oeis

A364132 — a(n) is the smallest positive integer such that from the set {1, 2, ..., a(n)} one can choose an increasing sequence (s(1), s(2), ..., s(n)) in which every segment has a unique sum of elements.1,2,4,5,7,10,12,13,15,18,21,24,25,29,30,33,36,38,41,47,50,52 A364153 — a(n) is the smallest positive integer such that from the set {1, 2, ..., a(n)} one can choose a sequence (s(1), s(2), ..., s(n)) in which every segment has a unique sum.1,2,3,5,6,7,9,10,12,13,14,17,18

status

open

evidence

formal

oeis

status

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