We say that $A \subseteq N$ has property $P$ if, for all $n \geq 1$ , there are only finitely many $a \in A$ such that $n + a$ is squarefree.We say that $A$ has property $Q$ if there are infinitely many $n$ such that $n + a$ is squarefree for all $a < n$ . How fast must sequences $A = {a_{1} < a_{2} < \dots}$ with properties $P$ or $Q$ increase?

Worked, still open.

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

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AMS 11 · solved (literature)

theorem erdos_1102.density_zero_of_P
    (A : ℕ → ℕ)
    (h_inc : StrictMono A)
    (hP : HasPropertyP (range A)) :
    Tendsto (fun j => (A j / j : ℝ)) atTop atTop

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