105751.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

9 declarations, 3 contain sorry, 0 declared as axioms

Imaginary part of $\prod_{k=0}^n (1 + k \cdot i)$, $i = \sqrt{-1}$

References:

A105751

Imports (1)

FormalConjecturesUtil

def OeisA105751.a The primary defining sequence a line 37 source
Source documentation
The primary defining sequence a. a n is the imaginary part of $\prod_{k=0}^n (1 + k \cdot i)$, where $i = \sqrt{-1}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: Complex, HAdd.hAdd, OfNat.ofNat, HMul.hMul, Complex.I, Int.floor, Real, Complex.im, Finset.prod, Finset.range
15 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℤ :=
…
```
theorem OeisA105751.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 43 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, Int, OeisA105751.a, OfNat.ofNat, Nat
Proof uses: id, of_eq_true, Complex, HAdd.hAdd, HMul.hMul, Complex.I, Int.floor, Real, Complex.im, Finset.prod, Finset.range, Eq.trans, True, congrFun', congrArg, Multiset.prod, Multiset.map, Finset.val, Singleton.singleton, Multiset and 16 more
36 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 0 := by
…
```
Mentions: OeisA105751.a
theorem OeisA105751.a_1 line 47 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA105751.a, OfNat.ofNat, Nat
Proof uses: id, of_eq_true, Complex, HAdd.hAdd, HMul.hMul, Complex.I, Int.floor, Real, Complex.im, Finset.prod, Finset.range, Eq.trans, True, congrFun', congrArg, Multiset.prod, Multiset.map, Finset.val, Singleton.singleton, Multiset and 27 more
43 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA105751.a_2 line 51 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA105751.a, OfNat.ofNat, Nat
Proof uses: id, of_eq_true, Complex, HAdd.hAdd, HMul.hMul, Complex.I, Int.floor, Real, Complex.im, Finset.prod, Finset.range, Eq.trans, True, congrFun', congrArg, Mathlib.Meta.NormNum.instAtLeastTwo, Complex.re, Multiset.prod, Multiset.map, Finset.val and 34 more
56 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA105751.a_3 line 55 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA105751.a, OfNat.ofNat, Nat
Proof uses: id, of_eq_true, Complex, HAdd.hAdd, HMul.hMul, Complex.I, Int.floor, Real, Complex.im, Finset.prod, Finset.range, Eq.trans, True, congrFun', congrArg, Complex.re, Mathlib.Meta.NormNum.instAtLeastTwo, Multiset.prod, Multiset.map, Finset.val and 44 more
79 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA105751.a_4 line 59 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA105751.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: id, of_eq_true, Complex, HAdd.hAdd, HMul.hMul, Complex.I, Int.floor, Real, Complex.im, Finset.prod, Finset.range, Eq.trans, True, inferInstance, Ring, congrFun', congrArg, Mathlib.Meta.NormNum.instAtLeastTwo, Complex.re, Multiset.prod and 45 more
82 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA105751.prime_divides_some_term Numerical calculation suggests that a similar division holds in this case sorry line 72 source
Source documentation
Numerical calculation suggests that a similar division holds in this case. Type 1: primes p that do not divide any element of the sequence {$a(n)$}. In this case, unlike in A105750, the set of type 1 primes is empty; that is, every prime p divides some term of this sequence.
Note: The triangular number n*(n+1)/2 divides $a(n)$ (see A164652). In particular, if $p$ is an odd prime then $p$ divides $a(p)$. For $p=2$, $2$ divides $a(4)=-40$.

depends on sorry what this means
Statement uses: Nat, Nat.Prime, Exists, And, Ne, Int, OeisA105751.a, OfNat.ofNat, Dvd.dvd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem prime_divides_some_term (p : ℕ) (hp : p.Prime) :
    ∃ n : ℕ, a n ≠ 0 ∧ (p : ℤ) ∣ a n := by
…
```
Mentions: OeisA105751.a
theorem OeisA105751.conjecture sorry line 83 source
depends on sorry what this means
Statement uses: Filter.Tendsto, Nat, Rat, HDiv.hDiv, HMul.hMul, OfNat.ofNat, padicValInt, OeisA105751.a, Filter.atTop, nhds
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA105751.conjecture.variants.moll_p_mod_4_eq_1 Moll's conjecture 5.5 extends to this sequence and takes the form: (ii) for the other primes of type sorry line 95 source
Source documentation
Moll's conjecture 5.5 extends to this sequence and takes the form: (ii) for the other primes of type $2$, the p-adic valuation $\nu_p(a(n)) \sim n/(p - 1)$ as $n \rightarrow \infty$.
(Type 2 primes consists of primes p == 1 (mod 4))

depends on sorry what this means
Statement uses: Nat, Nat.Prime, Eq, HMod.hMod, OfNat.ofNat, Filter.Tendsto, Rat, HDiv.hDiv, HMul.hMul, HSub.hSub, padicValInt, OeisA105751.a, Filter.atTop, nhds
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture.variants.moll_p_mod_4_eq_1 {p : ℕ} (hp : p.Prime) (h_mod : p % 4 = 1) :
    Tendsto (fun n ↦ ((p - 1 : ℚ) * (padicValInt p (a n) : ℚ)) / (n : ℚ)) atTop (nhds 1) := by
…
```
Mentions: OeisA105751.a