38107.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

Number of primes $< n^2$

Number of primes strictly less than $n^2$.

References:

A038107

Imports (1)

FormalConjecturesUtil

def OeisA38107.a Number of primes strictly less than $n^2$ line 33 source
Source documentation
Number of primes strictly less than $n^2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Nat.primesBelow, HPow.hPow, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ := (Nat.primesBelow (n ^ 2)).card
…
```
theorem OeisA38107.a_0 Value of the sequence a at 0 line 37 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA38107.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 a_0 : a 0 = 0 := by
…
```
Mentions: OeisA38107.a
theorem OeisA38107.a_1 Value of the sequence a at 1 line 42 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA38107.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 a_1 : a 1 = 0 := by
…
```
Mentions: OeisA38107.a
theorem OeisA38107.a_2 Value of the sequence a at 2 line 47 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA38107.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 a_2 : a 2 = 2 := by
…
```
Mentions: OeisA38107.a
theorem OeisA38107.a_3 Value of the sequence a at 3 line 52 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA38107.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 a_3 : a 3 = 4 := by
…
```
Mentions: OeisA38107.a
theorem OeisA38107.a_4 Value of the sequence a at 4 line 57 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA38107.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 a_4 : a 4 = 6 := by
…
```
Mentions: OeisA38107.a
theorem OeisA38107.a_5 Value of the sequence a at 5 line 62 source
Source documentation
Value of the sequence a at 5.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA38107.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 a_5 : a 5 = 9 := by
…
```
Mentions: OeisA38107.a
theorem OeisA38107.conjecture Conjecture: all the numbers $\sum_{i=j}^k \frac{1}{a(i)}$ with $1 < j \le k$ have pairwise distinct fractional parts sorry line 71 source
Source documentation
Conjecture: all the numbers $\sum_{i=j}^k \frac{1}{a(i)}$ with $1 < j \le k$ have pairwise distinct fractional parts.
- Zhi-Wei Sun, Sep 24 2015
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, LE.le, Eq, Real, Int.fract, Finset.sum, Finset.Icc, HDiv.hDiv, OeisA38107.a, And
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture (j k j' k' : ℕ)
    (hj : 1 < j) (hjk : j ≤ k) (hj' : 1 < j') (hj'k' : j' ≤ k')
    (h_eq : Int.fract (∑ i ∈ Finset.Icc j k, (1 : ℝ) / (a i : ℝ)) =
            Int.fract (∑ i ∈ Finset.Icc j' k', (1 : ℝ) / (a i : ℝ))) :
    j = j' ∧ k = k' := by
…
```
Mentions: OeisA38107.a