FormalConjectures/OEIS/109909.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

Conjectures associated with A109909

$a(n)$ = number of primes of the form $k(n-k)-1$.

References:

Imports (1)
  1. def OeisA109909.a $a(n)$ is the number of primes of the form $k(n-k)-1$ line 37 source

    Source documentation

    $a(n)$ is the number of primes of the form $k(n-k)-1$. This counts the number of distinct prime values in the set $\{k(n-k)-1 | 1 \le k < n\}$.

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

    Statement uses: Nat

    Proof uses: Finset.card, Finset.filter, Nat.Prime, Finset.image, HSub.hSub, HMul.hMul, OfNat.ofNat, Finset.Ico

    9 instances, coercions and auxiliary constants left out.

    def a (n : ℕ) : ℕ :=
…
  2. theorem OeisA109909.a_1 line 43 source

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

    Statement uses: Eq, Nat, OeisA109909.a, OfNat.ofNat

    Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl

    3 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  3. theorem OeisA109909.a_2 line 46 source

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

    Statement uses: Eq, Nat, OeisA109909.a, OfNat.ofNat

    Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl

    3 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  4. theorem OeisA109909.a_3 line 49 source

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

    Statement uses: Eq, Nat, OeisA109909.a, OfNat.ofNat

    Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl

    3 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  5. theorem OeisA109909.a_4 line 52 source

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

    Statement uses: Eq, Nat, OeisA109909.a, OfNat.ofNat

    Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl

    3 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  6. theorem OeisA109909.a_5 line 55 source

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

    Statement uses: Eq, Nat, OeisA109909.a, OfNat.ofNat

    Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl

    3 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  7. theorem OeisA109909.conjecture Conjecture: $a(n) > 0$ for $n > 3$ sorry line 61 source

    Source documentation

    Conjecture: $a(n) > 0$ for $n > 3$.

    depends on sorry what this means

    Statement uses: Nat, GT.gt, OfNat.ofNat, OeisA109909.a

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

    4 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem conjecture : ∀ n > 3, a n > 0 := by
…

    Mentions: OeisA109909.a