109227.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

7 declarations, 1 contain sorry, 0 declared as axioms

Conjectures associated with A109227

Binary strings that have 1's where primes occur, 0's elsewhere and every term ends with the $n$-th prime index.

Conjecture: $a(2)$ and $a(121)$ are primes. Are there any more?

References:

A109227

Imports (1)

FormalConjecturesUtil

def OeisA109227.a The primary defining sequence a line 43 source
Source documentation
The primary defining sequence a. $a(n)$ is the natural number whose decimal digits are given by the binary string of length $p_n$ where the $k$-th digit is $1$ if $k$ is prime and $0$ otherwise, with leading zeros dropped.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, Nat.nth, Nat.Prime, HSub.hSub, List, List.map, List.range, HAdd.hAdd, List.dropWhile, Decidable.decide, Nat.ofDigits, List.reverse
8 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA109227.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 54 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, OeisA109227.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 a_0 : a 0 = 0 := by
…
```
Mentions: OeisA109227.a
theorem OeisA109227.a_1 line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA109227.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Nat.ofDigits, List.reverse, List.dropWhile, Decidable.decide, List.map, ite, Nat.Prime, List.range, HAdd.hAdd, Nat.nth, congrArg, Nat.nth_prime_zero_eq_two, of_decide_eq_true, Bool, Bool.true, Eq.refl
7 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA109227.a_2 line 64 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA109227.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Nat.ofDigits, List.reverse, List.dropWhile, Decidable.decide, List.map, ite, Nat.Prime, List.range, HAdd.hAdd, Nat.nth, congrArg, Nat.nth_prime_one_eq_three, of_decide_eq_true, Bool, Bool.true, Eq.refl
7 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA109227.a_3 line 70 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA109227.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Nat.ofDigits, List.reverse, List.dropWhile, Decidable.decide, List.map, ite, Nat.Prime, List.range, HAdd.hAdd, Nat.nth, congrArg, Nat.nth_prime_two_eq_five, of_decide_eq_true, Bool, Bool.true, Eq.refl
7 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA109227.a_4 line 76 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA109227.a, OfNat.ofNat
Proof uses: id, Eq.mpr, Nat.ofDigits, List.reverse, List.dropWhile, Decidable.decide, List.map, ite, Nat.Prime, List.range, HAdd.hAdd, Nat.nth, congrArg, Nat.nth_prime_three_eq_seven, of_decide_eq_true, Bool, Bool.true, Eq.refl
7 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA109227.conjecture Conjecture: $a(2)$ and $a(121)$ are primes sorry line 85 source
Source documentation
Conjecture: $a(2)$ and $a(121)$ are primes. Are there any more?

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, GT.gt, OfNat.ofNat, Ne, Nat.Prime, OeisA109227.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 > 0, n ≠ 2 ∧ n ≠ 121 ∧ (a n).Prime := by
…
```
Mentions: OeisA109227.a