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Erdős problem 830

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We say that a,b∈Na,b\in \mathbb{N} are an amicable pair if σ(a)=σ(b)=a+b\sigma(a)=\sigma(b)=a+b. If A(x)A(x) counts the number of amicable 1≤a≤b≤x1\leq a\leq b\leq x then one can show that A(x)≤xexp⁡(−(12+o(1))(log⁡xlog⁡log⁡x)1/2)A(x) \leq x \exp(-(\tfrac{1}{2}+o(1))(\log x\log\log x)^{1/2}).

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830.lean

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We say that a,b∈Na,b\in \mathbb{N} are an amicable pair if σ(a)=σ(b)=a+b\sigma(a)=\sigma(b)=a+b. If A(x)A(x) counts the number of amicable 1≤a≤b≤x1\leq a\leq b\leq x then one can show that A(x)≤xexp⁡(−(log⁡x)1/3)A(x) \leq x \exp(-(\log x)^{1/3}).

FormalConjectures/ErdosProblems/830.leanErdos830.erdos_830.variants.pomerance1 lineExact file
∀ᶠ (x : ℝ) in Filter.atTop, Erdos830.A x ≤ x * Real.exp (-Real.nthRoot 3 (Real.log x))
SolvedStatement only, no proof