129365.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

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

11 declarations, 4 contain sorry, 0 declared as axioms

Ratio of product of GCDs to product of factorials of floor divisions

$$a(n) = \frac{\prod_{j=1}^n \prod_{k=1}^n \gcd(j,k)}{\prod_{k=1}^n (\lfloor n/k \rfloor!)^k}$$

References:

A129365

Imports (1)

FormalConjecturesUtil

def OeisA129365.a $a(n) = \frac{\prod_{j=1}^n \prod_{k=1}^n \gcd(j,k)}{\prod_{k=1}^n (\lfloor n/k \rfloor!)^k}$ line 34 source
Source documentation
$a(n) = \frac{\prod_{j=1}^n \prod_{k=1}^n \gcd(j,k)}{\prod_{k=1}^n (\lfloor n/k \rfloor!)^k}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Rat
Proof uses: Finset.prod, Finset.Icc, OfNat.ofNat, Nat.gcd, HPow.hPow, Nat.factorial, HDiv.hDiv
11 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℚ :=
…
```
def OeisA129365.b Sequence A004125: sum of remainders $n \bmod k$ for $1 \le k \le n$ line 40 source
Source documentation
Sequence A004125: sum of remainders $n \bmod k$ for $1 \le k \le n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.Icc, OfNat.ofNat, HMod.hMod
6 instances, coercions and auxiliary constants left out.
```
def b (n : ℕ) : ℕ :=
…
```
theorem OeisA129365.a_0 Value of the sequence a at 0 line 45 source
Source documentation
Value of the sequence a at 0.

extra axioms: _private.0.OeisA129365.a_0._native.decide.ax_1 what this means
Statement uses: Eq, Rat, OeisA129365.a, OfNat.ofNat, Nat
Proof uses: of_decide_eq_true
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 1 := by decide +native
…
```
Mentions: OeisA129365.a
theorem OeisA129365.a_1 Value of the sequence a at 1 line 49 source
Source documentation
Value of the sequence a at 1.

extra axioms: _private.0.OeisA129365.a_1._native.decide.ax_1 what this means
Statement uses: Eq, Rat, OeisA129365.a, OfNat.ofNat, Nat
Proof uses: of_decide_eq_true
6 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 = 1 := by decide +native
…
```
Mentions: OeisA129365.a
theorem OeisA129365.a_2 Value of the sequence a at 2 line 53 source
Source documentation
Value of the sequence a at 2.

extra axioms: _private.0.OeisA129365.a_2._native.decide.ax_1 what this means
Statement uses: Eq, Rat, OeisA129365.a, OfNat.ofNat, Nat
Proof uses: of_decide_eq_true
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 1 := by decide +native
…
```
Mentions: OeisA129365.a
theorem OeisA129365.a_3 Value of the sequence a at 3 line 57 source
Source documentation
Value of the sequence a at 3.

extra axioms: _private.0.OeisA129365.a_3._native.decide.ax_1 what this means
Statement uses: Eq, Rat, OeisA129365.a, OfNat.ofNat, Nat
Proof uses: of_decide_eq_true
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 1 := by decide +native
…
```
Mentions: OeisA129365.a
theorem OeisA129365.a_4 Value of the sequence a at 4 line 61 source
Source documentation
Value of the sequence a at 4.

extra axioms: _private.0.OeisA129365.a_4._native.decide.ax_1 what this means
Statement uses: Eq, Rat, OeisA129365.a, OfNat.ofNat, Nat
Proof uses: of_decide_eq_true
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 1 := by decide +native
…
```
Mentions: OeisA129365.a
theorem OeisA129365.conjecture1 sorry line 87 source
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Dvd.dvd, Finset.prod, Finset.Icc, HPow.hPow, Nat.factorial, HDiv.hDiv, Nat.gcd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA129365.conjecture2 sorry line 108 source
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Nat.Prime, Iff, Exists, And, Eq, Rat, OeisA129365.a, Dvd.dvd, LE.le, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA129365.conjecture3 sorry line 128 source
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Nat.Prime, Eq, Int, padicValRat, OeisA129365.a, HMul.hMul, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA129365.conjecture4 sorry line 145 source
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Nat.Prime, Eq, Int, padicValRat, OeisA129365.a, HMul.hMul, tsum, OeisA129365.b, HDiv.hDiv, HPow.hPow, SummationFilter.unconditional
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

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