102722.lean`

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

12 declarations, 1 contain sorry, 0 declared as axioms

Floor of sum of $\{n/k\}$

Given $n$, sum all division remainders $\{n/k\}$, with $k=1,\dots,n$. The value $a(n)$ is given by the floor of that sum. Note that $\{x\}:=x-[x]$. Conjecture: a(n) ~ (1-EulerGamma)n.

References:

A102722

Imports (1)

FormalConjecturesUtil

def OeisA102722.a The primary defining sequence a line 40 source
Source documentation
The primary defining sequence a. a n is the floor of the sum of all division remainders $\{n/k\}$, with $k=1,\dots,n$. $$a(n) = \left\lfloor \sum_{k=1}^n \left\{ \frac{n}{k} \right\} \right\rfloor$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Real, Finset.sum, Finset.Icc, OfNat.ofNat, Int.fract, HDiv.hDiv, Int.toNat, Int.floor
12 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
lemma OeisA102722.a_sum_1 line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Eq, Finset.sum, Finset.Icc, OfNat.ofNat
Proof uses: Finset, Singleton.singleton, rfl, Eq.mpr, id, congrArg, Finset.sum_singleton, Eq.refl
9 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma OeisA102722.a_sum_2 line 53 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Eq, Finset.sum, Finset.Icc, OfNat.ofNat, HAdd.hAdd
Proof uses: Finset, Insert.insert, Singleton.singleton, rfl, Eq.mpr, id, congrArg, Finset.sum_insert, of_decide_eq_true, Not, Membership.mem, Bool, Decidable.decide, Bool.true, Eq.refl, Finset.sum_singleton
23 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA102722.a_sum_3 line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Eq, Finset.sum, Finset.Icc, OfNat.ofNat, HAdd.hAdd
Proof uses: Finset, Insert.insert, Singleton.singleton, rfl, Eq.mpr, id, congrArg, Finset.sum_insert, of_decide_eq_true, Not, Membership.mem, Bool, Decidable.decide, Bool.true, Eq.refl, Finset.sum_singleton, add_assoc
25 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma OeisA102722.a_sum_4 line 64 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Eq, Finset.sum, Finset.Icc, OfNat.ofNat, HAdd.hAdd
Proof uses: Finset, Insert.insert, Singleton.singleton, rfl, Eq.mpr, id, congrArg, Finset.sum_insert, of_decide_eq_true, Not, Membership.mem, Bool, Decidable.decide, Bool.true, Eq.refl, Finset.sum_singleton
24 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma OeisA102722.a_sum_5 line 71 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Eq, Finset.sum, Finset.Icc, OfNat.ofNat, HAdd.hAdd
Proof uses: Finset, Insert.insert, Singleton.singleton, rfl, Eq.mpr, id, congrArg, Finset.sum_insert, of_decide_eq_true, Not, Membership.mem, Bool, Decidable.decide, Bool.true, Eq.refl, Finset.sum_singleton
24 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
theorem OeisA102722.a_1 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 85 source
Source documentation
Term theorems verifying the first few values of the sequence against the official OEIS b-file

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA102722.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Real, Finset.sum, Finset.Icc, Int.fract, HDiv.hDiv, Int.toNat, Int.floor, congrFun', congrArg, Int, OeisA102722.a_sum_1, of_eq_true, Eq.trans, True, Mathlib.Meta.NormNum.IsNat.to_eq, inferInstance, AddMonoidWithOne, Rat.isNat_intFloor and 14 more
35 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 eval_a
…
```
Mentions: OeisA102722.a
theorem OeisA102722.a_2 line 88 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA102722.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Real, Finset.sum, Finset.Icc, Int.fract, HDiv.hDiv, Int.toNat, Int.floor, HAdd.hAdd, congrFun', congrArg, Int, OeisA102722.a_sum_2, of_eq_true, Eq.trans, True, Mathlib.Meta.NormNum.IsNat.to_eq, inferInstance, AddMonoidWithOne and 18 more
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA102722.a_3 line 91 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA102722.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Real, Finset.sum, Finset.Icc, Int.fract, HDiv.hDiv, Int.toNat, Int.floor, HAdd.hAdd, congrFun', congrArg, Int, OeisA102722.a_sum_3, of_eq_true, Eq.trans, True, Mathlib.Meta.NormNum.IsNat.to_eq, inferInstance, AddMonoidWithOne and 19 more
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA102722.a_4 line 94 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA102722.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Real, Finset.sum, Finset.Icc, Int.fract, HDiv.hDiv, Int.toNat, Int.floor, HAdd.hAdd, congrFun', congrArg, Int, OeisA102722.a_sum_4, of_eq_true, Eq.trans, True, Mathlib.Meta.NormNum.IsNat.to_eq, inferInstance, AddMonoidWithOne and 20 more
41 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA102722.a_5 line 97 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA102722.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Real, Finset.sum, Finset.Icc, Int.fract, HDiv.hDiv, Int.toNat, Int.floor, HAdd.hAdd, congrFun', congrArg, Int, OeisA102722.a_sum_5, of_eq_true, Eq.trans, True, Mathlib.Meta.NormNum.IsNat.to_eq, inferInstance, AddMonoidWithOne and 19 more
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA102722.conjecture sorry line 103 source
depends on sorry what this means
Statement uses: Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, OeisA102722.a, HMul.hMul, HSub.hSub, OfNat.ofNat, Real.eulerMascheroniConstant
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
29 instances, coercions and auxiliary constants left out.

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