If $ω_{k} (n)$ counts the number of distinct prime factors of $n$ which are $> k$ , then is it true that, for every $k \geq 1$ , $n \to \infty lim inf 0 \leq i < k \sum ω_{k} (n + i) \leq k ?$ Is it true that $n \to \infty lim sup (0 \leq i < k \sum ω (n + i)) \frac{lo g lo g n}{lo g n} = 1,$ where $ω$ counts the number of distinct prime factors without restriction?

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

Let [ S_k(n):=\sum_{i=0}^{k-1}\omega(n+i), \qquad\text{where }\omega(m)=|\\{p:\ p\mid m\\}|. ]

llm-hunter · gpt pro 5.2 · unverified

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

AMS 11 · open (literature)

theorem erdos_890.parts.a :
    answer(sorry) ↔
    ∀ k ≥ 1, liminf (fun n ↦ (∑ i ∈ range k, (ω_gt k (n + i) : EReal))) atTop ≤ k

open