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import FormalConjecturesUtilRecurrence involving LCM: $a(n) = x(n+1)/x(n) - 2$
$a(n) = x(n+1)/x(n) - 2$ where $x(1)=1$ and $x(n) = 2 x(n-1) + \operatorname{lcm}(x(n-1),n)$ for $n > 1$.
References:
namespace OeisA135508Auxiliary sequence $x(n)$ with $x(1)=1$ and $x(n) = 2 x(n-1) + \operatorname{lcm}(x(n-1),n)$.
def x : ℕ → ℕ
| 0 => 0
| 1 => 1
| n + 1 => 2 * x n + Nat.lcm (x n) (n + 1)$a(n) = x(n+1)/x(n) - 2$ for $n \ge 1$, and $a(0) = 0$.
def a (n : ℕ) : ℕ :=
if n = 0 then 0 else (x (n + 1) / x n) - 2
Value of the sequence a at 0.
@[category test, AMS 11]
theorem a_0 : a 0 = 0 := ⊢ a 0 = 0 All goals completed! 🐙
Value of the sequence a at 1.
@[category test, AMS 11]
theorem a_1 : a 1 = 2 := ⊢ a 1 = 2 All goals completed! 🐙
Value of the sequence a at 2.
@[category test, AMS 11]
theorem a_2 : a 2 = 3 := ⊢ a 2 = 3 All goals completed! 🐙
Value of the sequence a at 3.
@[category test, AMS 11]
theorem a_3 : a 3 = 1 := ⊢ a 3 = 1 All goals completed! 🐙
Value of the sequence a at 4.
@[category test, AMS 11]
theorem a_4 : a 4 = 1 := ⊢ a 4 = 1 All goals completed! 🐙Conjecture: For prime $p$ such that $p-2$ is not a prime, $a(p-1) = p$.
Bill McEachen, Sep 26 2025
@[category research open, AMS 11]
theorem conjecture (p : ℕ) (hp : p.Prime) (hp_twin : ¬ (p - 2).Prime) : a (p - 1) = p := p:ℕhp:Nat.Prime php_twin:¬Nat.Prime (p - 2)⊢ a (p - 1) = p
All goals completed! 🐙end OeisA135508