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declared open · formalizederdos:18
∀ᶠ (n : ℕ) in Filter.atTop, Erdos18.practicalH n.factorial < n
number theorydivisorsfactorialsA0051532 sources
declared open · formalizederdos:373
Erdos373.S.Finite
number theoryfactorialsA0031352 sources
declared open · formalizederdos:390
Let f(n)f(n) be the least mm for which n!n! can be written as a1⋯aka_1\cdots a_k with n<a1<⋯<ak=mn < a_1 < \cdots < a_k = m - the smallest possible largest factor in a factorization of n!n! into distinct integers all exceeding nn. Erdős, Guy and Selfridge proved f(n)−2n≍n/log⁡nf(n) - 2n \asymp n/\log n. Erdős asked whether there is a constant cc with f(n)−2n∼c nlog⁡n,f(n) - 2n \sim c\,\frac{n}{\log n}, and what it is. This preprint answers yes and names the constant: lim⁡n→∞(f(n)−2n)log⁡nn=402963959825970038185≈0.15516.\lim_{n\to\infty}\frac{(f(n)-2n)\log n}{n} = \frac{4029639598}{25970038185} \approx 0.15516.
number theoryfactorialsA1934294 sources
declared openerdos:393
Erdős problem 393
number theoryfactorialsA3883022 sources
declared open · formalizederdos:400
True ↔ ∀ k ≥ 2, ∃ c, Asymptotics.IsEquivalent Filter.atTop (fun x => ∑ n ∈ Finset.Icc 1 x, ↑(Erdos400.g k n)) fun x => c * ↑x * Real.log ↑x
number theoryfactorialspossible3 sources
declared openerdos:404
Erdős problem 404
number theoryfactorialsN/A
declared openerdos:478
Erdős problem 478
number theoryfactorialsA210184
declared open · formalizederdos:727
True ↔ ∀ k ≥ 2, {n | (n + k).factorial ^ 2 ∣ (2 * n).factorial}.Infinite
number theoryfactorialsA002503 · A343507 · A3893962 sources
declared open · formalizederdos:912
∃ c > 0, Asymptotics.IsEquivalent Filter.atTop (fun n => ↑(Erdos912.h n)) fun n => c * (↑n / Real.log ↑n) ^ (1 / 2)
number theoryfactorialsA0716262 sources
declared open · formalizederdos:1108
True ↔ ∀ k ≥ 2, {a | a ∈ Erdos1108.FactorialSums ∧ ∃ m, m ^ k = a}.Finite
number theoryfactorialsA051761 · A115645 · A0254942 sources

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