FormalConjectures/OEIS/1157.lean

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Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL
Lean4 Domain Explanation Language

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

7 declarations, 1 contain sorry, 0 declared as axioms

Sum of squares of divisors of $n$

References:

Imports (1)
  1. def OeisA1157.a a n is the sum of squares of divisors of $n$ line 32 source

    Source documentation

    a n is the sum of squares of divisors of $n$.

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Nat

    Proof uses: Finset.sum, Nat.divisors, HPow.hPow, OfNat.ofNat

    6 instances, coercions and auxiliary constants left out.

    def a (n : ℕ) : ℕ :=
…
  2. theorem OeisA1157.a_1 line 36 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA1157.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.

  3. theorem OeisA1157.a_2 line 39 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA1157.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.

  4. theorem OeisA1157.a_3 line 42 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA1157.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.

  5. theorem OeisA1157.a_4 line 45 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA1157.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.

  6. theorem OeisA1157.a_5 line 48 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA1157.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.

  7. theorem OeisA1157.conjecture Conjecture: For each k = 2,3,..., all the rational numbers $\frac{\sigma_k(n)}{n^k} = \sum_{d|n} \frac{1}{d^k}$ (n = 1,2,3,...) have pairwise distinct fractional parts sorry line 58 source

    Source documentation

    Conjecture: For each k = 2,3,..., all the rational numbers $\frac{\sigma_k(n)}{n^k} = \sum_{d|n} \frac{1}{d^k}$ (n = 1,2,3,...) have pairwise distinct fractional parts. - Zhi-Wei Sun, Oct 15 2015

    depends on sorry what this means

    Statement uses: Nat, LE.le, OfNat.ofNat, LT.lt, Ne, Rat, Int.fract, HDiv.hDiv, ArithmeticFunction, ArithmeticFunction.sigma, HPow.hPow

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    34 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem conjecture :
  ∀ k : ℕ, 2 ≤ k →
    ∀ n₁ n₂ : ℕ, 0 < n₁ → 0 < n₂ → n₁ ≠ n₂ →
      Int.fract (↑((sigma k) n₁) / (↑n₁ ^ k : ℚ)) ≠
        Int.fract (↑((sigma k) n₂) / (↑n₂ ^ k : ℚ)) := by
…