FormalConjectures.ErdosProblems.«469»

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import FormalConjecturesUtil

Erdős Problem 469

Reference: erdosproblems.com/469


namespace Erdos469

The proposition that n is a sum of distinct proper divisors.

def Nat.IsSumDivisors (n : ℕ) : Prop :=
  ∃ S ⊆ n.properDivisors, ∑ d ∈ S, d = n


open Erdos469

Let $A$ be the set of all $n$ such that $n = d_1 + ⋯ + d_k$ with $d_i$ distinct proper divisors of $n$, but this is not true for any $m ∣ n$ with $m < n$. Does: $$ \sum_{n ∈ A} \frac 1 n $$ converge?

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_469 :
    letI A := {n : ℕ | 0 < n ∧ n.IsSumDivisors ∧ ∀ m < n, m ∣ n → ¬ m.IsSumDivisors}
    answer(sorry) ↔ Summable fun n : A ↦ 1 / (n : ℝ) := by⊢ True ↔ Summable fun n => 1 / ↑↑n
  sorryAll goals completed! 🐙


end Erdos469