erdos_1064.variants.general_function

For any function $f(n)=o(n)$, we have $\phi(n)>\phi(n-\phi(n))+f(n)$ for almost all $n$. Reference: [LuPo02] Luca, Florian and Pomerance, Carl, On some problems of {M}\polhk akowski-{S}chinzel and {E}rd\H os concerning the arithmetical functions {$\phi$} and {$\sigma$}. Colloq. Math. (2002), 111--130.

theorem erdos_1064.variants.general_function (f : ℕ → ℕ)
    (hf : (fun n ↦ (f n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ))) :
    {n : ℕ | φ (n - φ n) + f n < φ n}.HasDensity 1

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