53067.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

9 declarations, 1 contain sorry, 0 declared as axioms

Concatenation of the next $n$ numbers

$a(n)$ is the concatenation of the next $n$ numbers: the integers from $\frac{(n-1)n}{2} + 1$ up to $\frac{n(n+1)}{2}$.

References:

A053067

Imports (1)

FormalConjecturesUtil

def OeisA53067.triangular The $n$-th triangular number, $T_n = \frac{n(n+1)}{2}$ line 35 source
Source documentation
The $n$-th triangular number, $T_n = \frac{n(n+1)}{2}$.

no axioms what this means
Statement uses: Nat
Proof uses: HDiv.hDiv, HMul.hMul, HAdd.hAdd, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.
```
def triangular (n : ℕ) : ℕ := n * (n + 1) / 2
```
def OeisA53067.concatenateNats Concatenates two natural numbers $a$ and $b$ base 10 line 38 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: OeisA53067.a
def OeisA53067.a $a(n)$ is the concatenation of the next $n$ numbers line 42 source
Source documentation
$a(n)$ is the concatenation of the next $n$ numbers.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, HAdd.hAdd, OeisA53067.triangular, HSub.hSub, List, List.Ico, List.foldl, OeisA53067.concatenateNats
6 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA53067.a_1 line 51 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA53067.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA53067.a_2 line 55 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA53067.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA53067.a_3 line 59 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA53067.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA53067.a_4 line 63 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA53067.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA53067.a_5 line 67 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA53067.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA53067.conjecture "The second term is a prime sorry line 76 source
Source documentation
"The second term is a prime. When is the next prime, if there is another?
- N. J. A. Sloane, Dec 16 2016"
depends on sorry what this means
Statement uses: Eq, Option, Nat, sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, dite, Exists, And, LT.lt, Nat.Prime, OeisA53067.a, Classical.propDecidable, Option.some, InfSet.sInf, Set.ofPred and 2 more
6 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) =
      if h : ∃ n, 2 < n ∧ (a n).Prime then
        some (sInf {n | 2 < n ∧ (a n).Prime})
      else
        none := by
…
```
Mentions: OeisA53067.a