87207.lean`

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L4DEL: Lean4 Domain Explanation Language

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

7 declarations, 1 contain sorry, 0 declared as axioms

Binary representation of primes that divide a number, in decimal

The value $a(n)$ is given by $$ a(n) = \sum_{p \mid n, p \text{ prime}} 2^{\pi(p) - 1} $$ where $\pi(p) = \mathrm{primeCounting}(p)$ gives the 1-based index of the prime $p$.

References:

A087207

Imports (1)

FormalConjecturesUtil

def OeisA87207.a Binary representation of the prime factors of $n$, represented in decimal: $a(n) = \sum_{p \in \mathrm{support}(\mathrm{factorization}(n))} 2^{\pi(p) - 1}$ line 37 source
Source documentation
Binary representation of the prime factors of $n$, represented in decimal: $a(n) = \sum_{p \in \mathrm{support}(\mathrm{factorization}(n))} 2^{\pi(p) - 1}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finsupp.support, Nat.factorization, HPow.hPow, OfNat.ofNat, HSub.hSub, Nat.primeCounting
10 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA87207.a_0 Value of the sequence a at 0 line 42 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA87207.a, OfNat.ofNat
2 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 decide +kernel
…
```
Mentions: OeisA87207.a
theorem OeisA87207.a_1 Value of the sequence a at 1 line 46 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA87207.a, OfNat.ofNat
2 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 decide +kernel
…
```
Mentions: OeisA87207.a
theorem OeisA87207.a_2 Value of the sequence a at 2 line 50 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA87207.a, OfNat.ofNat
2 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 = 1 := by decide +kernel
…
```
Mentions: OeisA87207.a
theorem OeisA87207.a_3 Value of the sequence a at 3 line 54 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA87207.a, OfNat.ofNat
2 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 = 2 := by decide +kernel
…
```
Mentions: OeisA87207.a
theorem OeisA87207.a_4 Value of the sequence a at 4 line 58 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA87207.a, OfNat.ofNat
2 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 = 1 := by decide +kernel
…
```
Mentions: OeisA87207.a
theorem OeisA87207.conjecture Starting at any $n$ and iterating the map $n \mapsto a(n)$, we will always reach $0$ sorry line 65 source
Source documentation
Starting at any $n$ and iterating the map $n \mapsto a(n)$, we will always reach $0$.
- Antti Karttunen, Jun 18,20 2017
depends on sorry what this means
Statement uses: Nat, Exists, Eq, Nat.iterate, OeisA87207.a, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture (n : ℕ) : ∃ k : ℕ, (a^[k]) n = 0 := by
…
```
Mentions: OeisA87207.a