108569.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

9 declarations, 1 contain sorry, 0 declared as axioms

Numbers $n$ such that $\phi(n) = \phi(n + \phi(n))$

References:

A108569

Imports (1)

FormalConjecturesUtil

def OeisA108569.A The predicate defining whether $k$ belongs to the sequence line 34 source
Source documentation
The predicate defining whether $k$ belongs to the sequence.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: And, LT.lt, OfNat.ofNat, Eq, Nat.totient, HAdd.hAdd
4 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
def A (k : ℕ) : Prop := 0 < k ∧ φ k = φ (k + φ k)
```
instance OeisA108569.(instance at line 36) line 36 source
def OeisA108569.a The primary defining sequence a line 44 source
Source documentation
The primary defining sequence a. a n is the $(n+1)$-th positive integer $k$ such that $\phi(k) = \phi(k + \phi(k))$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.nth, OeisA108569.A
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA108569.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 49 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, Nat, OeisA108569.a, OfNat.ofNat
Proof uses: OeisA108569.A, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.count, Nat.nth, Nat.nth_count, Eq.mp, congrArg
4 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 1 := by
…
```
Mentions: OeisA108569.a
theorem OeisA108569.a_1 line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108569.a, OfNat.ofNat
Proof uses: OeisA108569.A, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.count, Nat.nth, Nat.nth_count, Eq.mp, congrArg
4 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
theorem OeisA108569.a_2 line 63 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108569.a, OfNat.ofNat
Proof uses: OeisA108569.A, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.count, Nat.nth, Nat.nth_count, Eq.mp, congrArg
4 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
theorem OeisA108569.a_3 line 70 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108569.a, OfNat.ofNat
Proof uses: OeisA108569.A, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.count, Nat.nth, Nat.nth_count, Eq.mp, congrArg
4 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
theorem OeisA108569.a_4 line 77 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA108569.a, OfNat.ofNat
Proof uses: OeisA108569.A, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.count, Nat.nth, Nat.nth_count, Eq.mp, congrArg
4 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
theorem OeisA108569.conjecture Conjecture: Except for the first term all terms are even sorry line 85 source
Source documentation
Conjecture: Except for the first term all terms are even.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Even, OeisA108569.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture : ∀ n, 0 < n → Even (a n) := by
…
```
Mentions: OeisA108569.a