108864.lean`

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10 declarations, 1 contain sorry, 0 declared as axioms

Numbers $n$ such that the perfect deficiency of $n$ is $\le 10$.

The perfect deficiency of $n$ (A109883) is the remainder after greedily subtracting from $n$ its divisors in increasing order, skipping any divisor larger than the current remainder.

References:

Imports (1)

FormalConjecturesUtil

def OeisA108864.perfectDeficiency The perfect deficiency of $n$ (A109883): the remainder after greedily subtracting from $n$ its divisors in increasing order, skipping any divisor larger than the current remainder line 43 source
Source documentation
The perfect deficiency of $n$ (A109883): the remainder after greedily subtracting from $n$ its divisors in increasing order, skipping any divisor larger than the current remainder.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: List.foldl, ite, And, Dvd.dvd, LE.le, HSub.hSub, List.range, HAdd.hAdd, OfNat.ofNat
10 instances, coercions and auxiliary constants left out.
```
def perfectDeficiency (n : ℕ) : ℕ :=
…
```
def OeisA108864.A The condition for a number $n$ to be in the sequence line 50 source
Source documentation
The condition for a number $n$ to be in the sequence. It satisfies $0 < n$ and its perfect deficiency is $\le 10$.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: And, LT.lt, OfNat.ofNat, LE.le, OeisA108864.perfectDeficiency
3 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
def A (n : ℕ) : Prop :=
…
```
instance OeisA108864.(instance at line 53) line 53 source
def OeisA108864.a The primary defining sequence a line 61 source
Source documentation
The primary defining sequence a. $a(n)$ is the $n$-th number (0-indexed) such that its perfect deficiency is $\le 10$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.nth, OeisA108864.A
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA108864.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 66 source
Source documentation
Term theorems verifying the first few values of the sequence against the official OEIS b-file

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108864.a, OfNat.ofNat
Proof uses: OeisA108864.A, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.count, Nat.nth, Nat.nth_count, Eq.mp, congrArg
4 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 1 := by
…
```
Mentions: OeisA108864.a
theorem OeisA108864.a_1 line 73 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108864.a, OfNat.ofNat
Proof uses: OeisA108864.A, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.count, Nat.nth, Nat.nth_count, Eq.mp, congrArg
4 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
theorem OeisA108864.a_2 line 80 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108864.a, OfNat.ofNat
Proof uses: OeisA108864.A, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.count, Nat.nth, Nat.nth_count, Eq.mp, congrArg
4 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
theorem OeisA108864.a_3 line 87 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108864.a, OfNat.ofNat
Proof uses: OeisA108864.A, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.count, Nat.nth, Nat.nth_count, Eq.mp, congrArg
4 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
theorem OeisA108864.a_4 line 94 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108864.a, OfNat.ofNat
Proof uses: OeisA108864.A, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.count, Nat.nth, Nat.nth_count, Eq.mp, congrArg
4 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
theorem OeisA108864.conjecture Is $1155$ the last odd number in this sequence? sorry line 105 source
Source documentation
Is $1155$ the last odd number in this sequence? ($1155$ is the $59$th term starting from $1$, corresponding to $a(58) = 1155$).

depends on sorry what this means
Statement uses: Iff, True, Nat, GT.gt, OfNat.ofNat, Even, OeisA108864.a
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.
```
theorem conjecture :
    answer(sorry) ↔ ∀ n > 58, Even (a n) := by
…
```
Mentions: OeisA108864.a