114216.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

Largest odd divisor of $a(n-1) + \textrm{prime}(n)$

$a(0)=0$; thereafter $a(n)$ = largest odd divisor of $a(n-1) + \textrm{prime}(n)$.

References:

A114216

Imports (1)

FormalConjecturesUtil

def OeisA114216.a The primary defining sequence a line 37 source
Source documentation
The primary defining sequence a. $a(n)$ is the largest odd divisor of $a(n-1) + \textrm{prime}(n)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
2 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA114216.a_0 line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA114216.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA114216.a_1 line 52 source
extra axioms: _private.0.OeisA114216.a_1._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA114216.a, OfNat.ofNat
Proof uses: id, Eq.mpr, HDiv.hDiv, HAdd.hAdd, Nat.nth, Nat.Prime, HPow.hPow, padicValNat, congrArg, Nat.nth_prime_zero_eq_two, of_decide_eq_true
12 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA114216.a_2 line 58 source
extra axioms: _private.0.OeisA114216.a_1._native.native_decide.ax_1, _private.0.OeisA114216.a_2._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA114216.a, OfNat.ofNat
Proof uses: id, Eq.mpr, HDiv.hDiv, HAdd.hAdd, Nat.nth, Nat.Prime, HPow.hPow, padicValNat, congrArg, OeisA114216.a_1, Nat.nth_prime_one_eq_three, of_decide_eq_true
12 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA114216.a_3 line 64 source
extra axioms: _private.0.OeisA114216.a_1._native.native_decide.ax_1, _private.0.OeisA114216.a_2._native.native_decide.ax_1, _private.0.OeisA114216.a_3._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA114216.a, OfNat.ofNat
Proof uses: id, Eq.mpr, HDiv.hDiv, HAdd.hAdd, Nat.nth, Nat.Prime, HPow.hPow, padicValNat, congrArg, OeisA114216.a_2, Nat.nth_prime_two_eq_five, of_decide_eq_true
12 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA114216.a_4 line 70 source
extra axioms: _private.0.OeisA114216.a_1._native.native_decide.ax_1, _private.0.OeisA114216.a_2._native.native_decide.ax_1, _private.0.OeisA114216.a_3._native.native_decide.ax_1, _private.0.OeisA114216.a_4._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA114216.a, OfNat.ofNat
Proof uses: id, Eq.mpr, HDiv.hDiv, HAdd.hAdd, Nat.nth, Nat.Prime, HPow.hPow, padicValNat, congrArg, OeisA114216.a_3, Nat.nth_prime_three_eq_seven, of_decide_eq_true
12 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA114216.a_5 line 76 source
extra axioms: _private.0.OeisA114216.a_1._native.native_decide.ax_1, _private.0.OeisA114216.a_2._native.native_decide.ax_1, _private.0.OeisA114216.a_3._native.native_decide.ax_1, _private.0.OeisA114216.a_4._native.native_decide.ax_1, _private.0.OeisA114216.a_5._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA114216.a, OfNat.ofNat
Proof uses: id, Eq.mpr, HDiv.hDiv, HAdd.hAdd, Nat.nth, Nat.Prime, HPow.hPow, padicValNat, congrArg, OeisA114216.a_4, Nat.nth_prime_four_eq_eleven, of_decide_eq_true
12 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA114216.conjecture Is $a(33900)$ the last term equal to $1$? sorry line 85 source
Source documentation
Is $a(33900)$ the last term equal to $1$?

depends on sorry what this means
Statement uses: Iff, True, Nat, GT.gt, OfNat.ofNat, Ne, OeisA114216.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 > 33900, a n ≠ 1 := by
…
```
Mentions: OeisA114216.a