62567.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

10 declarations, 1 contain sorry, 0 declared as axioms

Reversible multiples $a(3^n) = 10^{3^{n-2}} - 1$ for powers of $3$

First multiple of $n$ whose reverse is also divisible by $n$, or 0 if no such multiple exists.

References:

A062567
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.

Imports (1)

FormalConjecturesUtil

def OeisA62567.reverseNat The number whose digits in base 10 are $n$'s digits reversed line 39 source
Source documentation
The number whose digits in base 10 are $n$'s digits reversed.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.ofDigits, OfNat.ofNat, List.reverse, Nat.digits
2 instances, coercions and auxiliary constants left out.
```
def reverseNat (k : ℕ) : ℕ :=
…
```
def OeisA62567.a First multiple of $n$ whose reverse is also divisible by $n$, or 0 if no such multiple exists line 47 source
Source documentation
First multiple of $n$ whose reverse is also divisible by $n$, or 0 if no such multiple exists.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, And, GT.gt, Dvd.dvd, OeisA62567.reverseNat, HMul.hMul, dite, Exists, Classical.propDecidable, Nat.find, Not
9 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
lemma OeisA62567.a_eq_self line 64 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, Dvd.dvd, OeisA62567.reverseNat, Eq, OeisA62567.a
Proof uses: id, Eq.mpr, ite, And, HMul.hMul, dite, Exists, Classical.propDecidable, Nat.find, Not, congrArg, if_neg, Exists.intro, And.intro, Decidable.byContradiction, one_mul, dif_pos, LT.lt, propext, Nat.find_eq_iff and 17 more
30 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
lemma OeisA62567.reverseNat_of_lt line 79 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Ne, OfNat.ofNat, LT.lt, Eq, OeisA62567.reverseNat
Proof uses: id, Eq.mpr, Nat.ofDigits, List.reverse, Nat.digits, List.cons, List.nil, congrArg, List, Nat.digits_of_lt, Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA62567.a_1 line 85 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA62567.a, OfNat.ofNat
Proof uses: OeisA62567.a_eq_self, Decidable.byContradiction, GT.gt, Not, Eq.mpr, Dvd.dvd, OeisA62567.reverseNat, id, congrArg, OeisA62567.reverseNat_of_lt, of_decide_eq_true, Ne, Bool, Decidable.decide, Bool.true, Eq.refl, LT.lt, dvd_refl
9 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma OeisA62567.a_2 line 89 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA62567.a, OfNat.ofNat
Proof uses: OeisA62567.a_eq_self, Decidable.byContradiction, GT.gt, Not, Eq.mpr, Dvd.dvd, OeisA62567.reverseNat, id, congrArg, OeisA62567.reverseNat_of_lt, of_decide_eq_true, Ne, Bool, Decidable.decide, Bool.true, Eq.refl, LT.lt, dvd_refl
9 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma OeisA62567.a_3 line 93 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA62567.a, OfNat.ofNat
Proof uses: OeisA62567.a_eq_self, Decidable.byContradiction, GT.gt, Not, Eq.mpr, Dvd.dvd, OeisA62567.reverseNat, id, congrArg, OeisA62567.reverseNat_of_lt, of_decide_eq_true, Ne, Bool, Decidable.decide, Bool.true, Eq.refl, LT.lt, dvd_refl
9 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma OeisA62567.a_4 line 97 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA62567.a, OfNat.ofNat
Proof uses: OeisA62567.a_eq_self, Decidable.byContradiction, GT.gt, Not, Eq.mpr, Dvd.dvd, OeisA62567.reverseNat, id, congrArg, OeisA62567.reverseNat_of_lt, of_decide_eq_true, Ne, Bool, Decidable.decide, Bool.true, Eq.refl, LT.lt, dvd_refl
9 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma OeisA62567.a_5 line 101 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA62567.a, OfNat.ofNat
Proof uses: OeisA62567.a_eq_self, Decidable.byContradiction, GT.gt, Not, Eq.mpr, Dvd.dvd, OeisA62567.reverseNat, id, congrArg, OeisA62567.reverseNat_of_lt, of_decide_eq_true, Ne, Bool, Decidable.decide, Bool.true, Eq.refl, LT.lt, dvd_refl
9 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem OeisA62567.a_three_pow_eq sorry line 115 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Iff, Eq, OeisA62567.a, HPow.hPow, HSub.hSub, Or
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.