109905.lean`

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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 A109905

$a(n)$ is the greatest prime of the form $k(n-k)+1$, where $k$ can take values from $1$ to $\lfloor n/2 \rfloor$. $a(n)=0$ if no such prime exists.

References:

A109905

Imports (1)

FormalConjecturesUtil

def OeisA109905.a $a(n)$ is the greatest prime of the form $k(n-k)+1$, where $k$ can take values from $1$ to $\lfloor n/2 \rfloor$ line 39 source
Source documentation
$a(n)$ is the greatest prime of the form $k(n-k)+1$, where $k$ can take values from $1$ to $\lfloor n/2 \rfloor$. $a(n) = 0$ if no such prime exists.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sup, Finset.filter, Nat.Prime, Finset.image, HAdd.hAdd, HMul.hMul, HSub.hSub, OfNat.ofNat, Finset.Icc, HDiv.hDiv, id
16 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA109905.a_1 line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA109905.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.
theorem OeisA109905.a_2 line 49 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA109905.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.
theorem OeisA109905.a_3 line 52 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA109905.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.
theorem OeisA109905.a_4 line 55 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA109905.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.
theorem OeisA109905.a_5 line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA109905.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.
theorem OeisA109905.conjecture $a(n) = 0$ for $n = 1$, $6$, $30$ and $54$ sorry line 64 source
Source documentation
$a(n) = 0$ for $n = 1$, $6$, $30$ and $54$. Are there any others?

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, GT.gt, OfNat.ofNat, Eq, OeisA109905.a, Not, Membership.mem, Finset, Insert.insert, Singleton.singleton
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 conjecture :
    answer(sorry) ↔ ∃ n : ℕ, n > 0 ∧ a n = 0 ∧ n ∉ ({1, 6, 30, 54} : Finset ℕ) := by
…
```
Mentions: OeisA109905.a