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import FormalConjecturesUtilErdős Problem 359
namespace Erdos359
open Filter Asymptotics
The predicate that A is monotone, A 0 = n and for all j, A (j + 1) is the smallest natural number that
cannot be written as a sum of consecutive terms of A 0, ..., A j
def IsGoodFor (A : ℕ → ℕ) (n : ℕ) : Prop := A 0 = n ∧ StrictMono A ∧
∀ j, IsLeast
{m : ℕ | A j < m ∧ ∀ a b, Finset.Icc a b ⊆ Finset.Iic j → m ≠ ∑ i ∈ Finset.Icc a b, A i}
(A <| j + 1)Let $a_1< a_2 < ⋯ $ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s. Show that $a_k / k \to \infty$.
@[category research open, AMS 11]
theorem erdos_359.parts.i (A : ℕ → ℕ) (hA : IsGoodFor A 1) :
atTop.Tendsto (fun k ↦ (A k : ℝ) / k) atTop := A:ℕ → ℕhA:IsGoodFor A 1⊢ Tendsto (fun k => ↑(A k) / ↑k) atTop atTop
All goals completed! 🐙Let $a_1< a_2 < ⋯ $ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s. Show that $a_k / k ^ {1 + c} \to 0$ for any $c > 0$.
@[category research open, AMS 11]
theorem erdos_359.parts.ii (A : ℕ → ℕ) (hA : IsGoodFor A 1) (c : ℝ) (hc : 0 < c):
atTop.Tendsto (fun k ↦ A k / (k : ℝ) ^ (1 + c)) (nhds 0) := A:ℕ → ℕhA:IsGoodFor A 1c:ℝhc:0 < c⊢ Tendsto (fun k => ↑(A k) / ↑k ^ (1 + c)) atTop (nhds 0)
All goals completed! 🐙
Suppose monotone sequence $A$ satisfies the following: A 0 = 1 and for all j, A (j + 1) is the
smallest natural number that cannot be written as a sum of consecutive terms of A 0, ..., A j.
Then the first few terms of $A$ are $1,2,4,5,8,10,14,15,...$.
@[category test, AMS 11]
theorem erdos_359.variants.isGoodFor_1_low_values (A : ℕ → ℕ) (hA : IsGoodFor A 1) :
A '' (Set.Iic 7) = {1, 2, 4, 5, 8, 10, 14, 15} := A:ℕ → ℕhA:IsGoodFor A 1⊢ A '' Set.Iic 7 = {1, 2, 4, 5, 8, 10, 14, 15}
All goals completed! 🐙
Suppose monotone sequence $A$ satisfies the following: A 0 = 1 and for all j, A (j + 1) is the
smallest natural number that cannot be written as a sum of consecutive terms of A 0, ..., A j.
Then it is conjectured that $$a_k ~ \frac{k \log k}{\log \log k}$$.
@[category research open, AMS 11]
theorem erdos_359.variants.isGoodFor_1_asymptotic (A : ℕ → ℕ) (hA : IsGoodFor A 1) :
(fun k ↦ (A k : ℝ)) ~[atTop] (fun k ↦ k * (k : ℝ).log / (k : ℝ).log.log) := A:ℕ → ℕhA:IsGoodFor A 1⊢ (fun k => ↑(A k)) ~[atTop] fun k => ↑k * Real.log ↑k / Real.log (Real.log ↑k)
All goals completed! 🐙
end Erdos359