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import FormalConjecturesUtilErdős Problem 1054
Reference: erdosproblems.com/1054
namespace Erdos1054open Filter AsymptoticsLet $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$.
noncomputable def f (n : ℕ) : ℕ :=
open scoped Classical in
if h : ∃ᵉ (m) (k ≥ 1), n = ∑ i < k, Nat.nth (· ∈ m.divisors) i then
Nat.find h
else 0Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$?
@[category research open, AMS 11]
theorem erdos_1054.parts.i : answer(sorry) ↔ (fun n ↦ (f n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ)) := ⊢ True ↔ (fun n ↦ ↑(f n)) =o[atTop] fun n ↦ ↑n
All goals completed! 🐙Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$ for almost all $n$?
@[category research open, AMS 11]
theorem erdos_1054.parts.ii : answer(sorry) ↔ ∃ (A : Set ℕ), A.HasDensity 1 ∧
(fun (n : A) ↦ (f ↑n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ)) := ⊢ True ↔ ∃ A, A.HasDensity 1 ∧ (fun n ↦ ↑(f ↑n)) =o[atTop] fun n ↦ ↑↑n
All goals completed! 🐙Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $\limsup f(n)/n=\infty$?
@[category research open, AMS 11]
theorem erdos_1054.parts.iii : answer(sorry) ↔ ∃ (A : Set ℕ), A.HasDensity 1 ∧
atTop.limsup (fun n ↦ (f n : EReal) / n) = ⊤ := ⊢ True ↔ ∃ A, A.HasDensity 1 ∧ limsup (fun n ↦ ↑(f n) / ↑n) atTop = ⊤
All goals completed! 🐙Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Show that $f$ is undefined at $n=2$, i.e. we get the junk value $0$.
m:ℕk:ℕhk:k ≥ 1hsum:2 = 1 + ∑ x ∈ (Finset.Iio k).erase 0, Nat.nth (fun x ↦ x ∈ m.divisors) xhm:m ≠ 0hk0:0 ∈ Finset.Iio ki:ℕhi_mem:i ∈ (Finset.Iio k).erase 0hi_ne:Nat.nth (fun x ↦ x ∈ m.divisors) i ≠ 0⊢ False
m:ℕk:ℕhk:k ≥ 1hsum:2 = 1 + ∑ x ∈ (Finset.Iio k).erase 0, Nat.nth (fun x ↦ x ∈ m.divisors) xhm:m ≠ 0hk0:0 ∈ Finset.Iio ki:ℕhi_mem:i ∈ (Finset.Iio k).erase 0hi_ne:Nat.nth (fun x ↦ x ∈ m.divisors) i ≠ 0h2:2 ≤ Nat.nth (fun x ↦ x ∈ m.divisors) i⊢ False
m:ℕk:ℕhk:k ≥ 1hsum:2 = 1 + ∑ x ∈ (Finset.Iio k).erase 0, Nat.nth (fun x ↦ x ∈ m.divisors) xhm:m ≠ 0hk0:0 ∈ Finset.Iio ki:ℕhi_mem:i ∈ (Finset.Iio k).erase 0hi_ne:Nat.nth (fun x ↦ x ∈ m.divisors) i ≠ 0h2:2 ≤ Nat.nth (fun x ↦ x ∈ m.divisors) ithis:2 ≤ ∑ x ∈ (Finset.Iio k).erase 0, Nat.nth (fun x ↦ x ∈ m.divisors) x⊢ False
All goals completed! 🐙Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Show that $f$ is undefined at $n=5$, i.e. we get the junk value $0$.
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x ↦ x ∈ m.divisorshsum:5 = ∑ i < k, Nat.nth p ihpdef:p = fun x ↦ x ∈ m.divisorshfin:(Set.ofPred p).Finitehg0:Nat.nth p 0 = 1hlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p jrefute4:Nat.nth p 1 ≠ 4hk3:3 ≤ khg2:Nat.nth p 2 ≠ 0hc3:2 < hfin.toFinset.cardhg1:1 + 1 ≤ Nat.nth p 1hg2':2 + 1 ≤ Nat.nth p 2hsub:1 + Nat.nth p 1 + Nat.nth p 2 ≤ ∑ i < k, Nat.nth p i⊢ False
All goals completed! 🐙end Erdos1054