145355.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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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

6 declarations, 1 contain sorry, 0 declared as axioms

Factorial distance to nearest square

The sequence is defined as $$a(n) = \mathrm{round}\left(\frac{\mathrm{round}(\sqrt{n!})}{\left|(\mathrm{round}(\sqrt{n!}))^2 - n!\right|}\right)$$ for $n \ge 2$.

References:

A145355

Imports (1)

FormalConjecturesUtil

def OeisA145355.a The sequence $a(n) = \mathrm{round}(\mathrm{round}(\sqrt{n!})/|(\mathrm{round}(\sqrt{n!}))^2 - n!|)$ line 40 source
Source documentation
The sequence $a(n) = \mathrm{round}(\mathrm{round}(\sqrt{n!})/|(\mathrm{round}(\sqrt{n!}))^2 - n!|)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Real, Nat.factorial, Int, round, Real.sqrt, abs, HSub.hSub, HPow.hPow, OfNat.ofNat, Int.toNat, HDiv.hDiv
19 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA145355.a_2 Value of the sequence a at 2 line 49 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA145355.a, OfNat.ofNat
Proof uses: id, of_eq_true, Real, Nat.factorial, Int, round, Real.sqrt, abs, HSub.hSub, HPow.hPow, Int.toNat, HDiv.hDiv, Eq.trans, True, congrFun', congrArg, Int.floor, HAdd.hAdd, congr, Mathlib.Meta.NormNum.instAtLeastTwo and 62 more
116 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 1 := by
…
```
Mentions: OeisA145355.a
theorem OeisA145355.a_3 Value of the sequence a at 3 line 56 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA145355.a, OfNat.ofNat
Proof uses: id, of_eq_true, Real, Nat.factorial, Int, round, Real.sqrt, abs, HSub.hSub, HPow.hPow, Int.toNat, HDiv.hDiv, Eq.trans, True, congrFun', congrArg, Int.toNat.match_1, Int.floor, HAdd.hAdd, Mathlib.Meta.NormNum.instAddMonoidWithOne' and 64 more
111 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 1 := by
…
```
Mentions: OeisA145355.a
theorem OeisA145355.a_4 Value of the sequence a at 4 line 63 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA145355.a, OfNat.ofNat
Proof uses: id, of_eq_true, Real, Nat.factorial, Int, round, Real.sqrt, abs, HSub.hSub, HPow.hPow, Int.toNat, HDiv.hDiv, Eq.trans, True, congrFun', congrArg, Int.toNat.match_1, congr, Mathlib.Meta.NormNum.instAtLeastTwo, Mathlib.Meta.NormNum.IsNat.to_eq and 50 more
105 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 5 := by
…
```
Mentions: OeisA145355.a
theorem OeisA145355.a_5 Value of the sequence a at 5 line 71 source
Source documentation
Value of the sequence a at 5.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA145355.a, OfNat.ofNat
Proof uses: id, of_eq_true, Real, Nat.factorial, Int, round, Real.sqrt, abs, HSub.hSub, HPow.hPow, Int.toNat, HDiv.hDiv, Eq.trans, True, congrFun', congrArg, Int.toNat.match_1, congr, Int.floor, HAdd.hAdd and 57 more
104 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem a_5 : a 5 = 11 := by
…
```
Mentions: OeisA145355.a
theorem OeisA145355.conjecture This sequence suggests that the distance between a factorial and the closest power is tightly bounded sorry line 81 source
Source documentation
This sequence suggests that the distance between a factorial and the closest power is tightly bounded.

depends on sorry what this means
Statement uses: Exists, Nat, LE.le, OfNat.ofNat, OeisA145355.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 : ∃ C : ℕ, ∀ n : ℕ, 2 ≤ n → a n ≤ C := by
…
```
Mentions: OeisA145355.a