51293.lean`

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

8 declarations, 1 contain sorry, 0 declared as axioms

Asymptotics of subsets of $\{1, 2, \dots, n\}$ with integer average

Number of nonempty subsets of $\{1, 2, 3, \dots, n\}$ whose elements have an integer average.

Benoit Cloitre conjectured the asymptotic expansion: $$a(n) = \frac{2^{n+1}}{n} \left(1 + \frac{1}{n} + \frac{3}{n^2} + \frac{13}{n^3} + \frac{75}{n^4} + \frac{541}{n^5} + o\left(\frac{1}{n^5}\right)\right)$$ where the coefficients $1, 1, 3, 13, 75, 541, \dots$ are the Fubini numbers (preferential arrangements, A000670).

References:

A051293
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
A000670

Imports (1)

FormalConjecturesUtil

def OeisA51293.a Number of nonempty subsets of $\{1, 2, 3, \dots, n\}$ whose elements have an integer average line 49 source
Source documentation
Number of nonempty subsets of $\{1, 2, 3, \dots, n\}$ whose elements have an integer average.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Finset, Finset.filter, And, Finset.Nonempty, Dvd.dvd, Finset.sum, id, Finset.powerset, Finset.Icc, OfNat.ofNat
8 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
def OeisA51293.aReal line 55 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: OeisA51293.a
2 instances, coercions and auxiliary constants left out.
lemma OeisA51293.a_1 line 59 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA51293.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.
lemma OeisA51293.a_2 line 62 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA51293.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.
lemma OeisA51293.a_3 line 65 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA51293.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.
lemma OeisA51293.a_4 line 68 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA51293.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.
lemma OeisA51293.a_5 line 71 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA51293.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 OeisA51293.tendsto_aReal_asymptotic sorry line 82 source
depends on sorry what this means
Statement uses: Filter.Tendsto, Nat, Real, HDiv.hDiv, HSub.hSub, OeisA51293.aReal, HMul.hMul, HPow.hPow, OfNat.ofNat, HAdd.hAdd, Filter.atTop, nhds
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
56 instances, coercions and auxiliary constants left out.

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