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module
public import FormalConjecturesUtil$a(1)=1, a(2)=1, a(n)=(2^{a(n-1)} + 1)/a(n-2)$
The sequence is defined by $a(1) = 1, a(2) = 1$, and for $n \ge 3$, $$a(n) = \frac{2^{a(n-1)} + 1}{a(n-2)}$$
References:
@[expose] public sectionnamespace OeisA78590$a(1)=1, a(2)=1, a(n)=(2^{a(n-1)} + 1)/a(n-2)$.
def a : ℕ → ℕ
| 0 => 0
| 1 => 1
| 2 => 1
| n + 3 => (2 ^ a (n + 2) + 1) / a (n + 1)
Value of the sequence a at 1.
@[category test, AMS 11]
theorem a_1 : a 1 = 1 := ⊢ a 1 = 1
All goals completed! 🐙
Value of the sequence a at 2.
@[category test, AMS 11]
theorem a_2 : a 2 = 1 := ⊢ a 2 = 1
All goals completed! 🐙
Value of the sequence a at 3.
@[category test, AMS 11]
theorem a_3 : a 3 = 3 := ⊢ a 3 = 3
All goals completed! 🐙
Value of the sequence a at 4.
@[category test, AMS 11]
theorem a_4 : a 4 = 9 := ⊢ a 4 = 9
All goals completed! 🐙
Value of the sequence a at 5.
@[category test, AMS 11]
theorem a_5 : a 5 = 171 := ⊢ a 5 = 171
All goals completed! 🐙Are all terms integers?
No. Exact divisibility first fails at $n = 7$; the corresponding rational term has reduced denominator $19$.
@[category research solved, AMS 11,
formal_proof using lean4 at
"https://github.com/KitaKen1/oeis-a078590-counterexample/blob/3bdf1cee8e7e4cbc4fe83354e4be8ca1eee46de3/lean/OeisA78590CounterexampleFC.lean#L55-L61"]
theorem conjecture :
answer(False) ↔ ∀ n : ℕ, 3 ≤ n → a (n - 2) ∣ 2 ^ a (n - 1) + 1 := ⊢ False ↔ ∀ (n : ℕ), 3 ≤ n → OeisA78590.a (n - 2) ∣ 2 ^ OeisA78590.a (n - 1) + 1
All goals completed! 🐙end OeisA78590