Erdős #489 (?, AMS 11). STATEMENT: The set of positive integers not divisible by any element of `A`. FORMAL: erdos_489: answer(sorry) ↔ ∀ (A : Set ℕ), (fun x : ℕ => (((Finset.Icc 1 x).filter (· ∈ A)).card : ℝ)) =o[atTop] (fun x : ℕ => (x : ℝ).sqrt) → (sievedSet A).Infinite → ∃ L : ℝ, Tendsto (fun x : ℕ => GapSumSq A x / (x : ℝ)) atTop (𝓝 L); erdos_489: ∃ L : ℝ, Tendsto (fun x : ℕ => GapSumSq {n | ∃ p, Nat. KNOWN/REMARKS: # Erdős Problem 489 *Reference:* erdosproblems.com/489 | The squared-gap sum `∑_{b_i < x} (b_{i+1} - b_i)²`, where `b_i` enumerates the positive integers not divisible by any element of `A`. | Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap [1,x]\rvert=o(x^{1/2})$. Let $B=\{ n\geq 1 : a\nmid n\textrm{ for all }a\in A\}$. If $B=\{b_1 < b_2 < \cdots\}$ then is it true that $$\lim_{x \to RELATED: [145] OEIS: []

id

vpr_49792ded2d0d06ea

frontier

Erdős problems frontier

kind

finding.add

created

2026-06-04