Erdős Problem #890 — open problem in number theory

Erdős Problem #890

If ω_k(n) counts the number of distinct prime factors of n which are >k, then is it true that, for every k≥ 1, liminf_n→ ∞Σ_0≤ i < kω_k(n+i)≤ k?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-erdos-890,
  title        = {Erdős Problem #890},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-890}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

erdos_890.parts.a. If $ω_{k} (n)$ counts the number of distinct prime factors of $n$ which are $> k$ , then is it true that, for every $k \geq 1$ , $n \to \infty lim inf 0 \leq i < k \sum ω_{k} (n + i) \leq k ?$

erdos_890.parts.b. 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?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«890» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

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

theorem erdos_890.parts.b :
    answer(sorry) ↔ ∀ k ≥ 1, limsup (fun n ↦ (∑ i ∈ range k, (ω (n + i) : EReal)) *
      (log (log n) / log n)) atTop = 1

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/890. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/890
[ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.