87719.lean`

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

8 declarations, 2 contain sorry, 0 declared as axioms

Conjectures associated with A087719

Define $\varsigma(n)$ the smallest prime factor of $n$ (Nat.minFac). Let $a_n$ be the least number such that the count of numbers $k \le a_n$ with $k > \varsigma(k)^n$ exceeds the count of numbers with $k \le \varsigma(k)^n$.

The conjecture states that $a_n = 3^n + 3 \cdot 2^n + 6$ for $n \ge 1$.

References:

A087719

Imports (1)

FormalConjecturesUtil

def OeisA87719.countExceeding Count of numbers k in {1, ..., m} where k > (minFac k)^n line 40 source
Source documentation
Count of numbers k in {1, ..., m} where k > (minFac k)^n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Finset.filter, GT.gt, HPow.hPow, Nat.minFac, Finset.Icc, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.
```
def countExceeding (n m : ℕ) : ℕ :=
…
```
def OeisA87719.countNotExceeding Count of numbers k in {1, ..., m} where k ≤ (minFac k)^n line 44 source
Source documentation
Count of numbers k in {1, ..., m} where k ≤ (minFac k)^n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Finset.filter, LE.le, HPow.hPow, Nat.minFac, Finset.Icc, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.
```
def countNotExceeding (n m : ℕ) : ℕ :=
…
```
theorem OeisA87719.a_exists There exists m such that countExceeding n m > countNotExceeding n m sorry line 49 source
Source documentation
There exists m such that countExceeding n m > countNotExceeding n m.

depends on sorry what this means
Statement uses: Nat, Exists, GT.gt, OeisA87719.countExceeding, OeisA87719.countNotExceeding
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_exists (n : ℕ) : ∃ m, countExceeding n m > countNotExceeding n m := by
…
```
Mentions: OeisA87719.countExceeding, OeisA87719.countNotExceeding
def OeisA87719.a The sequence a(n): least m such that countExceeding n m > countNotExceeding n m line 53 source
Source documentation
The sequence a(n): least m such that countExceeding n m > countNotExceeding n m.

depends on sorry what this means
Statement uses: Nat
Proof uses: Nat.find, GT.gt, OeisA87719.countExceeding, OeisA87719.countNotExceeding, OeisA87719.a_exists
2 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA87719.a_1 a(1) = 15 line 58 source
Source documentation
a(1) = 15.

depends on sorry what this means
Statement uses: Eq, Nat, OeisA87719.a, OfNat.ofNat
Proof uses: Eq.mpr, Nat.find, GT.gt, OeisA87719.countExceeding, OeisA87719.countNotExceeding, OeisA87719.a_exists, id, congrArg, OeisA87719.a.eq_1, And, LT.lt, Not, propext, Nat.find_eq_iff, And.intro, of_decide_eq_true, dite, LE.le, Eq.symm, le_antisymm and 7 more
27 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 15 := by
…
```
Mentions: OeisA87719.a
theorem OeisA87719.a_2 a(2) = 27 line 66 source
Source documentation
a(2) = 27.

depends on sorry what this means
Statement uses: Eq, Nat, OeisA87719.a, OfNat.ofNat
Proof uses: Eq.mpr, Nat.find, GT.gt, OeisA87719.countExceeding, OeisA87719.countNotExceeding, OeisA87719.a_exists, id, congrArg, OeisA87719.a.eq_1, And, LT.lt, Not, propext, Nat.find_eq_iff, And.intro, of_decide_eq_true, dite, LE.le, Eq.symm, le_antisymm and 7 more
39 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 27 := by
…
```
Mentions: OeisA87719.a
theorem OeisA87719.a_3 a(3) = 57 line 74 source
Source documentation
a(3) = 57.

depends on sorry what this means
Statement uses: Eq, Nat, OeisA87719.a, OfNat.ofNat
Proof uses: Eq.mpr, Nat.find, GT.gt, OeisA87719.countExceeding, OeisA87719.countNotExceeding, OeisA87719.a_exists, id, congrArg, OeisA87719.a.eq_1, And, LT.lt, Not, propext, Nat.find_eq_iff, And.intro, of_decide_eq_true, dite, LE.le, Eq.symm, le_antisymm and 7 more
69 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 57 := by
…
```
Mentions: OeisA87719.a
theorem OeisA87719.a_formula sorry line 84 source
depends on sorry what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Eq, OeisA87719.a, HAdd.hAdd, HPow.hPow, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

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