76495.lean`

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

7 declarations, 1 contain sorry, 0 declared as axioms

Smallest $x$ such that $\sigma(x) \bmod x = n$

The sequence $a(n)$ is the smallest positive integer $x$ such that $\sigma_1(x) \bmod x = n$, or $0$ if no such $x$ exists.

References:

A076495

Imports (1)

FormalConjecturesUtil

def OeisA76495.a Smallest positive integer $x$ such that $\sigma_1(x) \bmod x = n$, or $0$ if no such $x$ exists line 39 source
Source documentation
Smallest positive integer $x$ such that $\sigma_1(x) \bmod x = n$, or $0$ if no such $x$ exists.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: dite, Exists, And, LT.lt, OfNat.ofNat, Eq, HMod.hMod, ArithmeticFunction, ArithmeticFunction.sigma, Classical.propDecidable, Nat.find, Not
11 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA76495.a_1 Value of the sequence a at 1 line 47 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA76495.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, HMod.hMod, ArithmeticFunction, ArithmeticFunction.sigma, Classical.propDecidable, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff, of_decide_eq_true, Bool, Decidable.decide and 10 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 2 := by
…
```
Mentions: OeisA76495.a
theorem OeisA76495.a_2 Value of the sequence a at 2 line 57 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA76495.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, HMod.hMod, ArithmeticFunction, ArithmeticFunction.sigma, Classical.propDecidable, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff, of_decide_eq_true, Bool, Decidable.decide and 10 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 20 := by
…
```
Mentions: OeisA76495.a
theorem OeisA76495.a_3 Value of the sequence a at 3 line 67 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA76495.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, HMod.hMod, ArithmeticFunction, ArithmeticFunction.sigma, Classical.propDecidable, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff, of_decide_eq_true, Bool, Decidable.decide and 10 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 4 := by
…
```
Mentions: OeisA76495.a
theorem OeisA76495.a_4 Value of the sequence a at 4 line 77 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA76495.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, HMod.hMod, ArithmeticFunction, ArithmeticFunction.sigma, Classical.propDecidable, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff, of_decide_eq_true, Bool, Decidable.decide and 10 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 9 := by
…
```
Mentions: OeisA76495.a
theorem OeisA76495.a_6 Value of the sequence a at 6 line 87 source
Source documentation
Value of the sequence a at 6.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA76495.a, OfNat.ofNat
Proof uses: id, dite, Exists, And, LT.lt, HMod.hMod, ArithmeticFunction, ArithmeticFunction.sigma, Classical.propDecidable, Nat.find, Not, Eq.mpr, congrFun', congrArg, dif_pos, propext, Nat.find_eq_iff, of_decide_eq_true, Bool, Decidable.decide and 10 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem a_6 : a 6 = 25 := by
…
```
Mentions: OeisA76495.a
theorem OeisA76495.conjecture At present, the 0 entry for $n = 5$ is only a conjecture sorry line 101 source
Source documentation
At present, the 0 entry for $n = 5$ is only a conjecture. That is, it is conjectured that there is no positive integer $x$ such that $\sigma_1(x) \bmod x = 5$.

depends on sorry what this means
Statement uses: Eq, Nat, OeisA76495.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 : a 5 = 0 := by
…
```
Mentions: OeisA76495.a