111291.lean`

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

11 declarations, 2 contain sorry, 0 declared as axioms

Number of refactorable numbers (A033950) $\le 10^n$

A number $k$ is refactorable if its number of divisors, $\tau(k)$, divides $k$.

References:

A111291
[Co99] Colton, S., Refactorable numbers - a machine invention. J. Integer Seq. 2 (1999), Article 99.1.2.
[Ze02] Zelinsky, J., Tau numbers: a partial proof of a conjecture and other results. J. Integer Seq. 5 (2002), Article 02.2.8.
[Sp85] Spiro, C., How often is the number of divisors of n a divisor of n? J. Number Theory 21 (1985), 81--100.

Imports (2)

def OeisA111291.countRefactorableNat Helper function: number of refactorable numbers $\le m$ line 43 source
Source documentation
Helper function: number of refactorable numbers $\le m$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Finset.filter, Dvd.dvd, Nat.divisors, Finset.Icc, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
def countRefactorableNat (m : ℕ) : ℕ :=
…
```
def OeisA111291.a a n is the number of refactorable numbers $\le 10^n$ line 50 source
Source documentation
a n is the number of refactorable numbers $\le 10^n$. A number $k$ is refactorable if its number of divisors, $\tau(k)$, divides $k$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: OeisA111291.countRefactorableNat, HPow.hPow, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA111291.a_0 line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA111291.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA111291.a_1 line 57 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA111291.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA111291.a_2 line 60 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA111291.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA111291.a_3 line 63 source
extra axioms: _private.0.OeisA111291.a_3._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA111291.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA111291.a_4 line 66 source
extra axioms: _private.0.OeisA111291.a_4._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA111291.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def OeisA111291.countRefactorable countRefactorable x is the number of refactorable numbers $\le x$ line 71 source
Source documentation
countRefactorable x is the number of refactorable numbers $\le x$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat
Proof uses: dite, GE.ge, OfNat.ofNat, OeisA111291.countRefactorableNat, Int.toNat, Int, Nat.floor, Not
13 instances, coercions and auxiliary constants left out.
```
noncomputable def countRefactorable (x : ℝ) : ℕ :=
…
```
theorem OeisA111291.conjecture Simon Colton conjectures that the number of refactorable numbers less than $x$ is at least $\frac{x}{2\log x}$ sorry line 88 source
Source documentation
Simon Colton conjectures that the number of refactorable numbers less than $x$ is at least $\frac{x}{2\log x}$. This is an asymptotic claim, so we state it for sufficiently large $x$.
In this form it is a theorem: Zelinsky [Ze02, Theorem 7] proved that for every $k$ the number of refactorable numbers $\le n$ exceeds $k \pi(n)$ for all sufficiently large $n$, and $\pi(n) \sim n / \log n$. It also follows from Spiro's asymptotic [Sp85], by which the count is $\frac{x}{\sqrt{\log x}} (\log \log x)^{-1 + o(1)}$. See colton_conjecture for the pointwise conjecture that remains open.

depends on sorry what this means
Statement uses: Filter.Eventually, Real, GE.ge, OeisA111291.countRefactorable, HDiv.hDiv, HMul.hMul, OfNat.ofNat, Nat, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture : ∀ᶠ x in Filter.atTop,
    (countRefactorable x : ℝ) ≥ x / (2 * Real.log x) := by
…
```
Mentions: OeisA111291.countRefactorable
theorem OeisA111291.colton_conjecture Documented declaration: colton_conjecture sorry line 101 source
Source documentation
Colton's conjecture [Co99] as stated by Zelinsky [Ze02]: for every $n$, the number of refactorable numbers $\le n$ is at least half the number of primes $\le n$, i.e. $\pi(n) \le 2\,T(n)$.
Zelinsky [Ze02] proved this for all sufficiently large $n$ (see conjecture), with an explicit bound of $7.42 \cdot 10^{13}$ beyond which it holds; the remaining range is open.

depends on sorry what this means
Statement uses: Nat, LE.le, Nat.primeCounting, HMul.hMul, OfNat.ofNat, OeisA111291.countRefactorableNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem colton_conjecture : ∀ n : ℕ, Nat.primeCounting n ≤ 2 * countRefactorableNat n := by
…
```
Mentions: OeisA111291.countRefactorableNat
theorem OeisA111291.colton_conjecture_le_500 Colton's conjecture holds for $n \le 500$ line 106 source
Source documentation
Colton's conjecture holds for $n \le 500$.

extra axioms: _private.0.OeisA111291.colton_conjecture_le_500._native.native_decide.ax_1 what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Nat.primeCounting, HMul.hMul, OeisA111291.countRefactorableNat
Proof uses: of_decide_eq_true
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem colton_conjecture_le_500 : ∀ n ≤ 500, Nat.primeCounting n ≤ 2 * countRefactorableNat n := by
…
```
Mentions: OeisA111291.countRefactorableNat