67857.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

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

7 declarations, 1 contain sorry, 0 declared as axioms

Sum of $a(k)/k!$ over divisors equals harmonic number

The sequence $a(n)$ satisfies $\sum_{k \mid n} \frac{a(k)}{k!} = \sum_{j=1}^n \frac{1}{j} = H_n$, where the sum on the left is over positive divisors $k$ of $n$. By Möbius inversion, $$a(n) = n! \sum_{d \mid n} \mu(n/d) H_d$$ where $H_d = \sum_{j=1}^d \frac{1}{j}$ is the $d$-th harmonic number.

References:

Imports (1)

FormalConjecturesUtil

def OeisA67857.a The sequence $a(n) = n! \sum_{d \mid n} \mu(n/d) H_d$ for $n \ge 1$, and $a(0) = 0$ line 41 source
Source documentation
The sequence $a(n) = n! \sum_{d \mid n} \mu(n/d) H_d$ for $n \ge 1$, and $a(0) = 0$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Rat
Proof uses: ite, Eq, OfNat.ofNat, HMul.hMul, Nat.factorial, Finset.sum, Nat.divisors, ArithmeticFunction, Int, ArithmeticFunction.moebius, HDiv.hDiv, harmonic
17 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℚ :=
…
```
theorem OeisA67857.a_1 Value of the sequence a at 1 line 49 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, OeisA67857.a, OfNat.ofNat, Nat
3 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
…
```
Mentions: OeisA67857.a
theorem OeisA67857.a_2 Value of the sequence a at 2 line 54 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, OeisA67857.a, OfNat.ofNat, Nat
3 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
…
```
Mentions: OeisA67857.a
theorem OeisA67857.a_3 Value of the sequence a at 3 line 59 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, OeisA67857.a, OfNat.ofNat, Nat
3 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 = 5 := by
…
```
Mentions: OeisA67857.a
theorem OeisA67857.a_4 Value of the sequence a at 4 line 64 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, OeisA67857.a, OfNat.ofNat, Nat
3 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 = 14 := by
…
```
Mentions: OeisA67857.a
theorem OeisA67857.a_5 Value of the sequence a at 5 line 69 source
Source documentation
Value of the sequence a at 5.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, OeisA67857.a, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_5 : a 5 = 154 := by
…
```
Mentions: OeisA67857.a
theorem OeisA67857.conjecture sorry line 85 source
depends on sorry what this means
Statement uses: Iff, False, Nat, LT.lt, OfNat.ofNat, Rat, OeisA67857.a, And, Odd, ArithmeticFunction, ArithmeticFunction.cardDistinctFactors, LE.le
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

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