105720.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

Triangular matchstick numbers in the class of prime numbers

$a(n) = \sum_{k = n}^{2n} p_k$, where $p_k$ is the $k$-th prime.

References:

A105720

Imports (1)

FormalConjecturesUtil

def OeisA105720.a The primary defining sequence a line 40 source
Source documentation
The primary defining sequence a. $a(n)$ is the sum of primes from the $n$-th prime to the $2n$-th prime, with $p_1=2$. For $n=0$, we define $a(0) = 0$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, Finset.sum, Finset.Ico, HSub.hSub, HMul.hMul, Nat.nth, Nat.Prime
9 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
lemma OeisA105720.nth_prime_five line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: Nat.count, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, 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.
lemma OeisA105720.nth_prime_six line 52 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: Nat.count, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, 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.
lemma OeisA105720.nth_prime_seven line 59 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: Nat.count, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, 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 OeisA105720.a_0 Value of the sequence a at 0 line 76 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105720.a, OfNat.ofNat
Proof uses: Eq.mpr, ite, Finset.sum, Finset.Ico, HSub.hSub, HMul.hMul, Nat.nth, Nat.Prime, id, congrArg, OeisA105720.a.eq_1, if_pos, rfl, Eq.refl
10 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: OeisA105720.a
theorem OeisA105720.a_1 Value of the sequence a at 1 line 81 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105720.a, OfNat.ofNat
Proof uses: Eq.mpr, ite, Finset.sum, Finset.Ico, HSub.hSub, HMul.hMul, Nat.nth, Nat.Prime, id, congrArg, OeisA105720.a.eq_1, if_neg, of_decide_eq_true, Not, Bool, Decidable.decide, Bool.true, Eq.refl, HAdd.hAdd, Finset.sum_Ico_succ_top and 15 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 5 := by
…
```
Mentions: OeisA105720.a
theorem OeisA105720.a_2 Value of the sequence a at 2 line 87 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105720.a, OfNat.ofNat
Proof uses: Eq.mpr, ite, Finset.sum, Finset.Ico, HSub.hSub, HMul.hMul, Nat.nth, Nat.Prime, id, congrArg, OeisA105720.a.eq_1, if_neg, of_decide_eq_true, Not, Bool, Decidable.decide, Bool.true, Eq.refl, HAdd.hAdd, Finset.sum_Ico_succ_top and 16 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 15 := by
…
```
Mentions: OeisA105720.a
theorem OeisA105720.a_3 Value of the sequence a at 3 line 93 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105720.a, OfNat.ofNat
Proof uses: Eq.mpr, ite, Finset.sum, Finset.Ico, HSub.hSub, HMul.hMul, Nat.nth, Nat.Prime, id, congrArg, OeisA105720.a.eq_1, if_neg, of_decide_eq_true, Not, Bool, Decidable.decide, Bool.true, Eq.refl, HAdd.hAdd, Finset.sum_Ico_succ_top and 17 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 36 := by
…
```
Mentions: OeisA105720.a
theorem OeisA105720.a_4 Value of the sequence a at 4 line 99 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105720.a, OfNat.ofNat
Proof uses: Eq.mpr, ite, Finset.sum, Finset.Ico, HSub.hSub, HMul.hMul, Nat.nth, Nat.Prime, id, congrArg, OeisA105720.a.eq_1, if_neg, of_decide_eq_true, Not, Bool, Decidable.decide, Bool.true, Eq.refl, HAdd.hAdd, Finset.sum_Ico_succ_top and 18 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 67 := by
…
```
Mentions: OeisA105720.a
theorem OeisA105720.conjecture Terms are squares at only(?) three values of $n = 3, 6, 4072$: corresponding terms are 6^2, 13^2, and 15735^2 sorry line 108 source
Source documentation
Terms are squares at only(?) three values of $n = 3, 6, 4072$: corresponding terms are 6^2, 13^2, and 15735^2.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Iff, IsSquare, OeisA105720.a, Or, Eq
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 : ℕ, 0 < n → (IsSquare (a n) ↔ (n = 3 ∨ n = 6 ∨ n = 4072)) := by
…
```
Mentions: OeisA105720.a