60841.lean`

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L4DEL: Lean4 Domain Explanation Language

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

10 declarations, 2 contain sorry, 0 declared as axioms

Numerator of $1/\det(M)$ for $M[i,j] = 1/\operatorname{lcm}(i,j)$

Numerator of $1/\det(M)$ where $M$ is the $n \times n$ matrix with $M[i,j] = 1/\operatorname{lcm}(i,j)$.

References:

A060841
T. Adamczewski, "OEIS Open: How many conjectures can language models turn into theorems?", arXiv:2608.11941 [cs.AI], 2026.

Imports (1)

FormalConjecturesUtil

def OeisA60841.lcmMatrix The $n \times n$ matrix with entry $(i,j)$ equal to $1/\operatorname{lcm}(i+1, j+1)$ over $\mathbb{Q}$ line 39 source
Source documentation
The $n \times n$ matrix with entry $(i,j)$ equal to $1/\operatorname{lcm}(i+1, j+1)$ over $\mathbb{Q}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Matrix, Fin, Rat
Proof uses: Equiv, Matrix.of, HDiv.hDiv, OfNat.ofNat, Nat.lcm, HAdd.hAdd
12 instances, coercions and auxiliary constants left out.
```
def lcmMatrix (n : ℕ) : Matrix (Fin n) (Fin n) ℚ :=
…
```
def OeisA60841.a Numerator of $1/\det(M)$ where $M$ is the $n \times n$ matrix with $M[i,j] = 1/\operatorname{lcm}(i+1,j+1)$ line 44 source
Source documentation
Numerator of $1/\det(M)$ where $M$ is the $n \times n$ matrix with $M[i,j] = 1/\operatorname{lcm}(i+1,j+1)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: Rat.num, Inv.inv, Rat, Matrix.det, Fin, OeisA60841.lcmMatrix
4 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℤ :=
…
```
theorem OeisA60841.a_1 line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA60841.a, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA60841.a_2 line 52 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA60841.a, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA60841.a_3 line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA60841.a, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA60841.a_4 line 60 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA60841.a, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA60841.a_5 line 64 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA60841.a, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def OeisA60841.integerDetN The exceptional values of $n$ where $1/\det(M)$ is conjectured to be an integer line 68 source
Source documentation
The exceptional values of $n$ where $1/\det(M)$ is conjectured to be an integer.

no axioms what this means
Statement uses: Set, Nat
Proof uses: Union.union, Set.Icc, OfNat.ofNat, Insert.insert, Singleton.singleton
5 instances, coercions and auxiliary constants left out.
```
def integerDetN : Set ℕ :=
…
```
theorem OeisA60841.conjecture1 "Conjecture: $1/\det(M)$ is an integer only for n: 1 to 34, 36 and 38." - _Robert G sorry line 76 source
Source documentation
"Conjecture: $1/\det(M)$ is an integer only for n: 1 to 34, 36 and 38." - Robert G. Wilson v, Aug 02 2015

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Iff, Eq, Rat.den, Inv.inv, Rat, Matrix.det, Fin, OeisA60841.lcmMatrix, Membership.mem, Set, OeisA60841.integerDetN
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture1 :
    ∀ n : ℕ, 1 ≤ n → (((lcmMatrix n).det)⁻¹.den = 1 ↔ n ∈ integerDetN) := by
…
```
Mentions: OeisA60841.lcmMatrix, OeisA60841.integerDetN
theorem OeisA60841.conjecture2 "All denominators are powers of two (A000079)." - _Robert G sorry line 90 source
Source documentation
"All denominators are powers of two (A000079)." - Robert G. Wilson v, Aug 02 2015
This is false: for $n = 1807$ the denominator of $1/\det(M)$ is divisible by $3$. See T. Adamczewski, OEIS Open: How many conjectures can language models turn into theorems?, arXiv:2608.11941. The Lean disproof there uses the closed form $1/\det(M) = \prod_{k=1}^n k^2/\varphi(k)$ from the OEIS entry: https://github.com/epoch-research/LeanOpenProblems-results/blob/fd09021e79869476ef83cda231312f1a2a89c8d7/runs/oeis-full-50usd-ant-j0j0g4uzligm1k41/oeis_60841_conjecture_0/Submission/Spec.lean#L272

L4DEL explanation
"All denominators are powers of two (A000079)." - Robert G. Wilson v, Aug 02 2015 This is false: for $n = 1807$ the denominator of $1/\det(M)$ is divisible by $3$. See T. Adamczewski, OEIS Open: How many conjectures can language models turn into theorems?, arXiv:2608.11941. The Lean disproof there uses the closed form $1/\det(M) = \prod_{k=1}^n k^2/\varphi(k)$ from the OEIS entry: https://github.com/epoch-research/LeanOpenProblems-results/blob/fd09021e79869476ef83cda231312f1a2a89c8d7/runs/oeis-full-50usd-ant-j0j0g4uzligm1k41/oeis_60841_conjecture_0/Submission/Spec.lean#L272
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Not, Nat, LE.le, OfNat.ofNat, Exists, Eq, Rat.den, Inv.inv, Rat, Matrix.det, Fin, OeisA60841.lcmMatrix, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2 :
    ¬ ∀ n : ℕ, 1 ≤ n → ∃ k : ℕ, ((lcmMatrix n).det)⁻¹.den = 2 ^ k := by
…
```
Mentions: OeisA60841.lcmMatrix