67599.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

8 declarations, 1 contain sorry, 0 declared as axioms

Decimal encoding of the prime factorization of $n$

If $n$ has prime factorization $p_1^{e_1} \dots p_r^{e_r}$ with $p_1 < \dots < p_r$, then its decimal encoding is $p_1 e_1 \dots p_r e_r$.

References:

A067599

Imports (1)

FormalConjecturesUtil

def OeisA67599.concatenateNats Concatenates two natural numbers $a$ and $b$ base 10 line 36 source
Source documentation
Concatenates two natural numbers $a$ and $b$ base 10.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HAdd.hAdd, HMul.hMul, HPow.hPow, OfNat.ofNat, List.length, Nat.digits
9 instances, coercions and auxiliary constants left out.
```
def concatenateNats (a b : ℕ) : ℕ :=
…
```
Mentions: OeisA67599.a
def OeisA67599.a Decimal encoding of the prime factorization of $n$ line 40 source
Source documentation
Decimal encoding of the prime factorization of $n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, LT.lt, OfNat.ofNat, List, List.dedup, Nat.primeFactorsList, List.flatMap, List.cons, List.count, List.nil, List.foldl, OeisA67599.concatenateNats
5 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA67599.a_2 line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA67599.a, OfNat.ofNat
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA67599.a_3 line 52 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA67599.a, OfNat.ofNat
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA67599.a_4 line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA67599.a, OfNat.ofNat
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA67599.a_5 line 60 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA67599.a, OfNat.ofNat
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA67599.a_6 line 64 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA67599.a, OfNat.ofNat
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA67599.conjecture "$a(31) = a(177147) = 311$ sorry line 72 source
Source documentation
"$a(31) = a(177147) = 311$. Is there any solution to $a(n) = n$?
- Franklin T. Adams-Watters, Dec 18 2006"
depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, LE.le, OfNat.ofNat, Eq, OeisA67599.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture :
    answer(sorry) ↔ ∃ n : ℕ, 2 ≤ n ∧ a n = n := by
…
```
Mentions: OeisA67599.a