1. FormalConjectures
  2. OEIS
  3. «108»
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Fractional parts of sums of reciprocals of Catalan numbers $C(n)$

Catalan numbers $C(n) = \frac{1}{n+1}\binom{2n}{n}$.

The sum $\sum_{i=j}^k \frac{1}{a(i)}$ of reciprocals of Catalan numbers.

References:

    A000108

    arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.

namespace OeisA108open Nat Real Finset

Catalan numbers $C(n) = \frac{1}{n+1}\binom{2n}{n}$.

def a (n : ℕ) : ℕ := (Nat.choose (2 * n) n) / (n + 1)def aRat (n : ℕ) : ℚ := (a n : ℚ)⁻¹

The sum $\sum_{i=j}^k \frac{1}{a(i)}$ of reciprocals of Catalan numbers.

def catalanReciprocalSum (j k : ℕ) : ℚ := (Finset.Icc j k).sum aRat

The index condition on $(j, k)$ from the conjecture: $0 < \min{2,k} \le j \le k$. Since j and k are natural numbers, $0 < \min{2,k}$ is equivalent to $1 \le k$.

def IndexCond (j k : ℕ) : Prop := 1 ≤ k ∧ min 2 k ≤ j ∧ j ≤ kopen Int (fract)

The fractional part of a rational number.

def fracPart (q : ℚ) : ℚ := fract q@[category test, AMS 11] lemma a_0 : a 0 = 1 := ⊢ a 0 = 1 All goals completed! 🐙@[category test, AMS 11] lemma a_1 : a 1 = 1 := ⊢ a 1 = 1 All goals completed! 🐙@[category test, AMS 11] lemma a_2 : a 2 = 2 := ⊢ a 2 = 2 All goals completed! 🐙@[category test, AMS 11] lemma a_3 : a 3 = 5 := ⊢ a 3 = 5 All goals completed! 🐙@[category test, AMS 11] lemma a_4 : a 4 = 14 := ⊢ a 4 = 14 All goals completed! 🐙

Conjecture: All the rational numbers $\sum_{i=j}^k \frac{1}{a(i)}$ with $0 < \min{2,k} \le j \le k$ have pairwise distinct fractional parts.

A formal proof has been found with the methods described in arxiv/2605.22763.

@[category research solved, AMS 11, formal_proof using formal_conjectures at "https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/108.wip.lean#L255"] theorem catalanReciprocalSum_fracPart_inj : ∀ ⦃j₁ k₁ j₂ k₂ : ℕ⦄, IndexCond j₁ k₁ → IndexCond j₂ k₂ → (j₁, k₁) ≠ (j₂, k₂) → fracPart (catalanReciprocalSum j₁ k₁) ≠ fracPart (catalanReciprocalSum j₂ k₂) := ⊢ ∀ ⦃j₁ k₁ j₂ k₂ : ℕ⦄, IndexCond j₁ k₁ → IndexCond j₂ k₂ → (j₁, k₁) ≠ (j₂, k₂) → fracPart (catalanReciprocalSum j₁ k₁) ≠ fracPart (catalanReciprocalSum j₂ k₂) All goals completed! 🐙end OeisA108