135508.lean`

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L4DEL: Lean4 Domain Explanation Language

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

8 declarations, 1 contain sorry, 0 declared as axioms

Recurrence 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:

A135508

Imports (1)

FormalConjecturesUtil

def OeisA135508.x Auxiliary sequence $x(n)$ with $x(1)=1$ and $x(n) = 2 x(n-1) + \operatorname{lcm}(x(n-1),n)$ line 34 source
Source documentation
Auxiliary sequence $x(n)$ with $x(1)=1$ and $x(n) = 2 x(n-1) + \operatorname{lcm}(x(n-1),n)$.

no axioms what this means
Statement uses: Nat
2 instances, coercions and auxiliary constants left out.
```
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 $a(0) = 0$. -/
```
Mentions: OeisA135508.a
def OeisA135508.a $a(n) = x(n+1)/x(n) - 2$ for $n \ge 1$, and $a(0) = 0$ line 40 source
Source documentation
$a(n) = x(n+1)/x(n) - 2$ for $n \ge 1$, and $a(0) = 0$.

no axioms what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, HSub.hSub, HDiv.hDiv, OeisA135508.x, HAdd.hAdd
8 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA135508.a_0 Value of the sequence a at 0 line 45 source
Source documentation
Value of the sequence a at 0.

no axioms what this means
Statement uses: Eq, Nat, OeisA135508.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 0 := by rfl
…
```
Mentions: OeisA135508.a
theorem OeisA135508.a_1 Value of the sequence a at 1 line 49 source
Source documentation
Value of the sequence a at 1.

no axioms what this means
Statement uses: Eq, Nat, OeisA135508.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 2 := by rfl
…
```
Mentions: OeisA135508.a
theorem OeisA135508.a_2 Value of the sequence a at 2 line 53 source
Source documentation
Value of the sequence a at 2.

no axioms what this means
Statement uses: Eq, Nat, OeisA135508.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 3 := by rfl
…
```
Mentions: OeisA135508.a
theorem OeisA135508.a_3 Value of the sequence a at 3 line 57 source
Source documentation
Value of the sequence a at 3.

no axioms what this means
Statement uses: Eq, Nat, OeisA135508.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 1 := by rfl
…
```
Mentions: OeisA135508.a
theorem OeisA135508.a_4 Value of the sequence a at 4 line 61 source
Source documentation
Value of the sequence a at 4.

no axioms what this means
Statement uses: Eq, Nat, OeisA135508.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 1 := by rfl
…
```
Mentions: OeisA135508.a
theorem OeisA135508.conjecture Conjecture: For prime $p$ such that $p-2$ is not a prime, $a(p-1) = p$ sorry line 68 source
Source documentation
Conjecture: For prime $p$ such that $p-2$ is not a prime, $a(p-1) = p$.
- Bill McEachen, Sep 26 2025
depends on sorry what this means
Statement uses: Nat, Nat.Prime, Not, HSub.hSub, OfNat.ofNat, Eq, OeisA135508.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture (p : ℕ) (hp : p.Prime) (hp_twin : ¬ (p - 2).Prime) : a (p - 1) = p := by
…
```
Mentions: OeisA135508.a