5153.lean`

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

7 declarations, 1 contain sorry, 0 declared as axioms

Practical numbers

A positive integer $n$ is called a practical number (or panarithmic number) if every positive integer $m \le n$ can be represented as a sum of distinct divisors of $n$.

References:

A005153

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/OEIS/17666.lean

def OeisA5153.A A positive integer $n$ is practical if every $m \le n$ can be represented as a sum of distinct divisors of $n$ line 36 source
Source documentation
A positive integer $n$ is practical if every $m \le n$ can be represented as a sum of distinct divisors of $n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, LT.lt, OfNat.ofNat, Nat.IsPractical
2 instances, coercions and auxiliary constants left out.
```
def A (n : ℕ) : Prop :=
…
```
theorem OeisA5153.a_1 $1$ is a practical number line 41 source
Source documentation
$1$ is a practical number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA5153.A, OfNat.ofNat, Nat
Proof uses: And.intro, LT.lt, Nat.IsPractical, of_decide_eq_true, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl, LE.le, dite, Membership.mem, Set, subsetSums, Finset, Nat.divisors, Exists.intro, And, Finset.sum and 25 more
30 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : A 1 := by
…
```
Mentions: OeisA5153.A
theorem OeisA5153.a_2 $2$ is a practical number line 49 source
Source documentation
$2$ is a practical number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA5153.A, OfNat.ofNat, Nat
Proof uses: And.intro, LT.lt, Nat.IsPractical, of_decide_eq_true, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl, LE.le, dite, Membership.mem, Set, subsetSums, Finset, Nat.divisors, Exists.intro, And, Finset.sum and 28 more
38 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : A 2 := by
…
```
Mentions: OeisA5153.A
theorem OeisA5153.a_4 $4$ is a practical number line 58 source
Source documentation
$4$ is a practical number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA5153.A, OfNat.ofNat, Nat
Proof uses: And.intro, LT.lt, Nat.IsPractical, of_decide_eq_true, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl, LE.le, Finset, Nat.divisors, Insert.insert, Singleton.singleton, dite, Membership.mem, Set, subsetSums, Exists.intro and 23 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 29 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : A 4 := by
…
```
Mentions: OeisA5153.A
theorem OeisA5153.a_6 $6$ is a practical number line 70 source
Source documentation
$6$ is a practical number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA5153.A, OfNat.ofNat, Nat
Proof uses: And.intro, LT.lt, Nat.IsPractical, of_decide_eq_true, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl, LE.le, Finset, Nat.divisors, Insert.insert, Singleton.singleton, dite, Membership.mem, Set, subsetSums, Exists.intro and 23 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 39 steps
Open to load the checked before-and-after goal states.
```
theorem a_6 : A 6 := by
…
```
Mentions: OeisA5153.A
theorem OeisA5153.a_8 $8$ is a practical number line 84 source
Source documentation
$8$ is a practical number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: OeisA5153.A, OfNat.ofNat, Nat
Proof uses: And.intro, LT.lt, Nat.IsPractical, of_decide_eq_true, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl, LE.le, Finset, Nat.divisors, Insert.insert, Singleton.singleton, dite, Membership.mem, Set, subsetSums, Exists.intro and 23 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 49 steps
Open to load the checked before-and-after goal states.
```
theorem a_8 : A 8 := by
…
```
Mentions: OeisA5153.A
theorem OeisA5153.conjecture Conjecture: every odd number, beginning with 3, is the sum of a prime number and a practical number sorry line 104 source
Source documentation
Conjecture: every odd number, beginning with 3, is the sum of a prime number and a practical number.
- Hal M. Switkay, Jan 28 2023
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Odd, Exists, And, Nat.Prime, OeisA5153.A, Eq, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 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) (hn : 3 ≤ n) (hodd : Odd n) :
    ∃ p q : ℕ, p.Prime ∧ A q ∧ n = p + q := by
…
```
Mentions: OeisA5153.A