1. FormalConjectures
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  3. «60841»
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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:

namespace OeisA60841

The $n \times n$ matrix with entry $(i,j)$ equal to $1/\operatorname{lcm}(i+1, j+1)$ over $\mathbb{Q}$.

def lcmMatrix (n : ℕ) : Matrix (Fin n) (Fin n) ℚ := Matrix.of fun i j : Fin n ↦ 1 / ((Nat.lcm (i.val + 1) (j.val + 1) : ℚ))

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

def a (n : ℕ) : ℤ := ((lcmMatrix n).det)⁻¹.num@[category test, AMS 11 15] theorem a_1 : a 1 = 1 := ⊢ a 1 = 1 All goals completed! 🐙@[category test, AMS 11 15] theorem a_2 : a 2 = 4 := ⊢ a 2 = 4 All goals completed! 🐙@[category test, AMS 11 15] theorem a_3 : a 3 = 18 := ⊢ a 3 = 18 All goals completed! 🐙@[category test, AMS 11 15] theorem a_4 : a 4 = 144 := ⊢ a 4 = 144 All goals completed! 🐙@[category test, AMS 11 15] theorem a_5 : a 5 = 900 := ⊢ a 5 = 900 All goals completed! 🐙

The exceptional values of $n$ where $1/\det(M)$ is conjectured to be an integer.

def integerDetN : Set ℕ := Set.Icc 1 34 ∪ {36, 38}

"Conjecture: $1/\det(M)$ is an integer only for n: 1 to 34, 36 and 38. All denominators are powers of two (A000079). - Robert G. Wilson v, Aug 02 2015"

@[category research open, AMS 11 15] theorem conjecture : (∀ n : ℕ, 1 ≤ n → (((lcmMatrix n).det)⁻¹.den = 1 ↔ n ∈ integerDetN)) ∧ (∀ n : ℕ, 1 ≤ n → ∃ k : ℕ, ((lcmMatrix n).det)⁻¹.den = 2 ^ k) := ⊢ (∀ (n : ℕ), 1 ≤ n → ((lcmMatrix n).det⁻¹.den = 1 ↔ n ∈ integerDetN)) ∧ ∀ (n : ℕ), 1 ≤ n → ∃ k, (lcmMatrix n).det⁻¹.den = 2 ^ k All goals completed! 🐙end OeisA60841