100474.lean`

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L4DEL: Lean4 Domain Explanation Language

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

18 declarations, 2 contain sorry, 0 declared as axioms

Conjectures associated with A100474

$a(1) = 1$; $a(n)$ is the smallest integer such that $a(n) + a(n-1)$ has the first $n$ distinct prime factors not used before in this construction.

References:

A100474

Imports (1)

FormalConjecturesUtil

def OeisA100474.triangular The $n$-th triangular number line 35 source
Source documentation
The $n$-th triangular number.

no axioms what this means
Statement uses: Nat
Proof uses: HDiv.hDiv, HMul.hMul, HAdd.hAdd, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.
```
def triangular (n : ℕ) : ℕ := n * (n + 1) / 2
```
def OeisA100474.a The primary defining sequence a line 38 source
Source documentation
The primary defining sequence a.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
2 instances, coercions and auxiliary constants left out.
```
noncomputable def a : ℕ → ℕ
  | 0 => 0
  | 1 => 1
  | n + 2 =>
    (Finset.Ico (triangular (n + 1) - 1) (triangular (n + 2) - 1)).prod (Nat.nth Nat.Prime) -
      a (n + 1)

@[category API, AMS 11]
```
Mentions: OeisA100474.triangular, AMS
lemma OeisA100474.nth_prime_five line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA100474.nth_prime_six line 51 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA100474.nth_prime_seven line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA100474.nth_prime_eight line 61 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA100474.nth_prime_nine line 66 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA100474.nth_prime_ten line 71 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA100474.nth_prime_eleven line 76 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA100474.nth_prime_twelve line 81 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA100474.nth_prime_thirteen line 86 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA100474.a_1 Value of the sequence a at 1 line 92 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA100474.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.
```
theorem a_1 : a 1 = 1 := by rfl
…
```
Mentions: OeisA100474.a
theorem OeisA100474.a_2 Value of the sequence a at 2 line 96 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA100474.a, OfNat.ofNat
Proof uses: id, Eq.mpr, HSub.hSub, Finset.prod, Finset.Ico, Nat.nth, Nat.Prime, HMul.hMul, congrArg, HAdd.hAdd, Finset.prod_Ico_succ_top, of_decide_eq_true, LE.le, Bool, Decidable.decide, Bool.true, Eq.refl, congrFun', Eq.trans, Finset.prod_congr and 7 more
22 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 5 := by
…
```
Mentions: OeisA100474.a
theorem OeisA100474.a_3 Value of the sequence a at 3 line 106 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA100474.a, OfNat.ofNat
Proof uses: id, Eq.mpr, HSub.hSub, Finset.prod, Finset.Ico, Nat.nth, Nat.Prime, congrArg, OeisA100474.a_2, HMul.hMul, HAdd.hAdd, Finset.prod_Ico_succ_top, of_decide_eq_true, LE.le, Bool, Decidable.decide, Bool.true, Eq.refl, congrFun', Eq.trans and 9 more
22 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 380 := by
…
```
Mentions: OeisA100474.a
theorem OeisA100474.a_4 Value of the sequence a at 4 line 118 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA100474.a, OfNat.ofNat
Proof uses: id, Eq.mpr, HSub.hSub, Finset.prod, Finset.Ico, Nat.nth, Nat.Prime, congrArg, OeisA100474.a_3, HMul.hMul, HAdd.hAdd, Finset.prod_Ico_succ_top, of_decide_eq_true, LE.le, Bool, Decidable.decide, Bool.true, Eq.refl, congrFun', Eq.trans and 10 more
22 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 96197 := by
…
```
Mentions: OeisA100474.a
theorem OeisA100474.a_5 Value of the sequence a at 5 line 131 source
Source documentation
Value of the sequence a at 5.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA100474.a, OfNat.ofNat
Proof uses: id, Eq.mpr, HSub.hSub, Finset.prod, Finset.Ico, Nat.nth, Nat.Prime, congrArg, OeisA100474.a_4, HMul.hMul, HAdd.hAdd, Finset.prod_Ico_succ_top, of_decide_eq_true, LE.le, Bool, Decidable.decide, Bool.true, Eq.refl, congrFun', Eq.trans and 11 more
22 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
```
theorem a_5 : a 5 = 58546472 := by
…
```
Mentions: OeisA100474.a
theorem OeisA100474.conjecture After $a(2) = 5$, is there another prime? sorry line 147 source
Source documentation
After $a(2) = 5$, is there another prime?

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, GT.gt, OfNat.ofNat, Nat.Prime, OeisA100474.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 : answer(sorry) ↔ ∃ n > 2, (a n).Prime := by
…
```
Mentions: OeisA100474.a
theorem OeisA100474.next_semiprime sorry line 156 source
depends on sorry what this means
Statement uses: Eq, Nat, OfNat.ofNat, OeisA100474.a, InfSet.sInf, Set.ofPred, And, LT.lt, Nat.IsSemiprime
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.