Let $f (n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k \geq 1$ .Is it true that $f (n) = o (n)$ ? Or is this true only for almost all $n$ , and $lim sup f (n) / n = \infty$ ?

Worked, still open.

number theory · open · formalized (Lean) · 0 attempts

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

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unverified AI candidates (2)

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

This question is listed as Erdős Problem #1054, and the “strong claim” (f(n)=o(n)) has been **disproved**. ([Erdős Problems][1])

llm-hunter · gpt pro 5.2 · unverified

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

AMS 11 · open (literature)

theorem erdos_1054.parts.i : answer(sorry) ↔ (fun n ↦ (f n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ))

n

D_{n}

d_{1}, d_{1} + d_{2}, d_{1} + d_{2} + d_{3}, \dots

1 < d_{1} < d_{2} < \dots

n

D_{n} \ \cup_{m < n} D_{m}

f (N)

n

N \in D_{n}

f (N) = o (N)

N

open