FormalConjectures.OEIS.«363102»

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import FormalConjecturesUtil

Primality of continued fraction denominators $a(n)$ for $n \ge 3$

Auxiliary sequence A051403, defined as $$\frac{(n+2) \sum_{k=0}^n k!}{2}$$

Denominator of the continued fraction $1/(2-3/(3-4/(4-5/(...(n-1)-n/(-2)))))$. The sequence is defined by the formula: $$a(n) = \frac{n^2 - 2}{\gcd(n^2 - 2, 2 \cdot A051403(n-3) + n \cdot A051403(n-4))}$$ The formula is valid for $n \ge 3$.

References:

A363102

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

A051403

namespace OeisA363102open Nat Finset

Auxiliary sequence A051403, defined as $$\frac{(n+2) \sum_{k=0}^n k!}{2}$$

def a051403 (n : ℕ) : ℕ :=
  let fact_sum := Finset.sum (range (n + 1)) (fun k => k.factorial)
  ((n + 2) * fact_sum) / 2

def a (n : ℕ) : ℕ :=
  let num : ℕ := n ^ 2 - 2
  let a051403_nm3 := a051403 (n - 3)
  let a051403_nm4 := a051403 (n - 4)
  let denom_arg := 2 * a051403_nm3 + n * a051403_nm4
  -- The subtraction n^2 - 2 is safe for n >= 3.
  num / Nat.gcd num denom_arg

@[category test, AMS 11]
lemma a_3 : a 3 = 7 := by⊢ a 3 = 7 rflAll goals completed! 🐙

@[category test, AMS 11]
lemma a_4 : a 4 = 7 := by⊢ a 4 = 7 rflAll goals completed! 🐙

@[category test, AMS 11]
lemma a_5 : a 5 = 23 := by⊢ a 5 = 23 rflAll goals completed! 🐙

@[category test, AMS 11]
lemma a_6 : a 6 = 17 := by⊢ a 6 = 17 rflAll goals completed! 🐙

@[category test, AMS 11]
lemma a_7 : a 7 = 47 := by⊢ a 7 = 47 rflAll goals completed! 🐙

Conjecture: The sequence contains only 1's and primes.

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/363102.wip.lean#L283"]
theorem a_eq_one_or_prime : ∀ n : ℕ, 3 ≤ n → a n = 1 ∨ Nat.Prime (a n) := by⊢ ∀ (n : ℕ), 3 ≤ n → OeisA363102.a n = 1 ∨ Nat.Prime (OeisA363102.a n)
    sorryAll goals completed! 🐙

end OeisA363102