110854.lean`

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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

12 declarations, 1 contain sorry, 0 declared as axioms

Conjectures associated with A110854

$a(n) = \mathrm{prime}(2n+2) - \mathrm{prime}(2n+1) - \mathrm{prime}(2n) + \mathrm{prime}(2n-1)$, where $\mathrm{prime}(k)$ is the $k$-th prime number.

References:

A110854

Imports (1)

FormalConjecturesUtil

def OeisA110854.a The primary defining sequence a line 40 source
Source documentation
The primary defining sequence a. $a(n)$ is $\mathrm{prime}(2n+2) - \mathrm{prime}(2n+1) - \mathrm{prime}(2n) + \mathrm{prime}(2n-1)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: Nat.nth, Nat.Prime, HSub.hSub, OfNat.ofNat, ite, Eq, HAdd.hAdd, HMul.hMul
13 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℤ :=
…
```
lemma OeisA110854.nth_prime_five line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA110854.nth_prime_six line 51 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA110854.nth_prime_seven line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA110854.nth_prime_eight line 61 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma OeisA110854.nth_prime_nine line 66 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Nat.nth, Nat.Prime, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl, Nat.nth_count
3 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA110854.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 72 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, Int, OeisA110854.a, OfNat.ofNat, Nat
Proof uses: 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 a_0 : a 0 = 0 := by
…
```
Mentions: OeisA110854.a
theorem OeisA110854.a_1 line 76 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA110854.a, OfNat.ofNat, Nat
Proof uses: id, of_eq_true, HAdd.hAdd, HSub.hSub, Nat.nth, Nat.Prime, Eq.trans, True, congrFun', congrArg, Neg.neg, congr, Nat.nth_prime_three_eq_seven, Nat.nth_prime_two_eq_five, Mathlib.Meta.NormNum.IsNat.to_eq, Mathlib.Meta.NormNum.instAddMonoidWithOne, Mathlib.Meta.NormNum.IsInt.to_isNat, Mathlib.Meta.NormNum.isInt_sub, Int.ofNat, Eq.refl and 11 more
28 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA110854.a_2 line 81 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA110854.a, OfNat.ofNat, Nat
Proof uses: id, Eq.mpr, HAdd.hAdd, HSub.hSub, Nat.nth, Nat.Prime, congrArg, OeisA110854.nth_prime_five, Nat.nth_prime_four_eq_eleven, Nat.nth_prime_three_eq_seven, Nat.nth_prime_two_eq_five, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.instAddMonoidWithOne, Mathlib.Meta.NormNum.IsInt.to_isNat, Mathlib.Meta.NormNum.isInt_add, Int.negOfNat, Int.ofNat, Eq.refl, Mathlib.Meta.NormNum.isInt_sub, Mathlib.Meta.NormNum.IsNat.to_isInt and 2 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA110854.a_3 line 88 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA110854.a, OfNat.ofNat, Nat
Proof uses: id, Eq.mpr, HAdd.hAdd, HSub.hSub, Nat.nth, Nat.Prime, congrArg, OeisA110854.nth_prime_seven, OeisA110854.nth_prime_six, OeisA110854.nth_prime_five, Nat.nth_prime_four_eq_eleven, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.instAddMonoidWithOne, Mathlib.Meta.NormNum.IsInt.to_isNat, Mathlib.Meta.NormNum.isInt_add, Int.negOfNat, Int.ofNat, Eq.refl, Mathlib.Meta.NormNum.isInt_sub, Mathlib.Meta.NormNum.IsNat.to_isInt and 2 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA110854.a_4 line 94 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA110854.a, OfNat.ofNat, Nat
Proof uses: id, Eq.mpr, HAdd.hAdd, HSub.hSub, Nat.nth, Nat.Prime, congrArg, OeisA110854.nth_prime_nine, OeisA110854.nth_prime_eight, OeisA110854.nth_prime_seven, OeisA110854.nth_prime_six, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.instAddMonoidWithOne, Mathlib.Meta.NormNum.IsInt.to_isNat, Mathlib.Meta.NormNum.isInt_add, Int.negOfNat, Int.ofNat, Eq.refl, Mathlib.Meta.NormNum.isInt_sub, Mathlib.Meta.NormNum.IsNat.to_isInt and 2 more
20 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem OeisA110854.conjecture Do the absolute values cover A004275? sorry line 106 source
Source documentation
Do the absolute values cover A004275? A004275 is $1$ together with the nonnegative even numbers. The conjecture asks whether every member of A004275 occurs as $|a(n)|$ for some term of the sequence.

depends on sorry what this means
Statement uses: Nat, Or, Eq, OfNat.ofNat, Even, Exists, And, GT.gt, Int.natAbs, OeisA110854.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 :
  ∀ d : ℕ, (d = 1 ∨ Even d) → ∃ n > 0, d = (a n).natAbs := by
…
```
Mentions: OeisA110854.a