110566.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

8 declarations, 1 contain sorry, 0 declared as axioms

$a(n) = \operatorname{lcm}\{1,2,\dots,n\}/\operatorname{denom}(H(n))$

References:

A110566

Imports (1)

FormalConjecturesUtil

def OeisA110566.a The primary defining sequence a line 37 source
Source documentation
The primary defining sequence a. $a(n) = \frac{\operatorname{lcm}_{k=1}^n k}{\operatorname{den}(H_n)}$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HDiv.hDiv, Finset.lcm, Finset.Icc, OfNat.ofNat, id, Rat.den, harmonic
8 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA110566.a_1 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 42 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, OeisA110566.a, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, HDiv.hDiv, Finset.lcm, Finset.Icc, id, Rat.den, harmonic, congr, normalize, Singleton.singleton, Finset, Finset.Icc_self, Finset.lcm_singleton, normalize_eq, Units, Rat and 9 more
55 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 simp [a]
…
```
Mentions: OeisA110566.a
theorem OeisA110566.a_2 line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA110566.a, OfNat.ofNat
Proof uses: Eq.mpr, HDiv.hDiv, Finset.lcm, Finset.Icc, id, congrFun', congrArg, Rat.den, harmonic, Eq.trans, Inv.inv, Rat, Int.natAbs, Rat.num, HAdd.hAdd, harmonic_succ, zero_add, Nat.cast_one, inv_one, Rat.ofNat_add_den and 9 more
40 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA110566.a_3 line 50 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA110566.a, OfNat.ofNat
Proof uses: Eq.mpr, HDiv.hDiv, Finset.lcm, Finset.Icc, id, Rat.den, HAdd.hAdd, Rat, Inv.inv, congrFun', congrArg, harmonic, Eq.trans, harmonic_succ, zero_add, Nat.cast_one, inv_one, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_ratDen, Int.ofNat and 15 more
45 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA110566.a_4 line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA110566.a, OfNat.ofNat
Proof uses: Eq.mpr, HDiv.hDiv, Finset.lcm, Finset.Icc, id, Rat.den, HAdd.hAdd, Rat, Inv.inv, congrFun', congrArg, harmonic, Eq.trans, harmonic_succ, zero_add, Nat.cast_one, inv_one, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_ratDen, Int.ofNat and 15 more
45 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA110566.a_5 line 62 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA110566.a, OfNat.ofNat
Proof uses: Eq.mpr, HDiv.hDiv, Finset.lcm, Finset.Icc, id, Rat.den, HAdd.hAdd, Rat, Inv.inv, congrFun', congrArg, harmonic, Eq.trans, harmonic_succ, zero_add, Nat.cast_one, inv_one, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_ratDen, Int.ofNat and 15 more
45 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA110566.a_6 line 68 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA110566.a, OfNat.ofNat
Proof uses: Eq.mpr, HDiv.hDiv, Finset.lcm, Finset.Icc, id, Rat.den, HAdd.hAdd, Rat, Inv.inv, congrFun', congrArg, harmonic, Eq.trans, harmonic_succ, zero_add, Nat.cast_one, inv_one, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.isNat_ratDen, Int.ofNat and 15 more
45 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA110566.conjecture It is conjectured that every odd number occurs in this sequence sorry line 78 source
Source documentation
It is conjectured that every odd number occurs in this sequence.

depends on sorry what this means
Statement uses: Nat, Odd, Exists, And, GT.gt, OfNat.ofNat, Eq, OeisA110566.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 :
  ∀ m : ℕ, Odd m → ∃ n > 0, a n = m := by
…
```
Mentions: OeisA110566.a