114137.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, 2 contain sorry, 0 declared as axioms

Difference between first odd semiprime $> 2^n$ and $2^n$

References:

A114137

Imports (1)

FormalConjecturesUtil

def OeisA114137.a The primary defining sequence a line 38 source
Source documentation
The primary defining sequence a. $a(n)$ is the difference between first odd semiprime > $2^n$ and $2^n$. $$a(n) = \min \{s \mid s > 2^n \text{ and } s \text{ is an odd semiprime}\} - 2^n$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HPow.hPow, OfNat.ofNat, Set, Set.ofPred, And, GT.gt, Nat.IsSemiprime, Odd, HSub.hSub, InfSet.sInf
10 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
lemma OeisA114137.a_eq_of line 44 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, And, Nat.IsSemiprime, Odd, LT.lt, HPow.hPow, OfNat.ofNat, Not, Eq, OeisA114137.a, HSub.hSub
Proof uses: id, InfSet.sInf, Set.ofPred, GT.gt, IsLeast.csInf_eq, And.intro, Membership.mem, Set, lowerBounds, LE.le, Decidable.byContradiction, Eq.mp, Eq.mpr, congrArg, Eq.refl
34 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
theorem OeisA114137.a_1 line 60 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA114137.a, OfNat.ofNat
Proof uses: OeisA114137.a_eq_of, of_eq_true, And, Nat.IsSemiprime, Odd, Eq.trans, True, congr, congrArg, Ne, ArithmeticFunction, ArithmeticFunction.cardFactors, eq_true, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, congrFun', List.length and 55 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
theorem OeisA114137.a_2 line 69 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA114137.a, OfNat.ofNat
Proof uses: OeisA114137.a_eq_of, of_eq_true, And, Nat.IsSemiprime, Odd, Eq.trans, True, congr, congrArg, Ne, ArithmeticFunction, ArithmeticFunction.cardFactors, eq_true, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, congrFun', List.length and 56 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
theorem OeisA114137.a_3 line 78 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA114137.a, OfNat.ofNat
Proof uses: OeisA114137.a_eq_of, of_eq_true, And, Nat.IsSemiprime, Odd, Eq.trans, True, congr, congrArg, Ne, ArithmeticFunction, ArithmeticFunction.cardFactors, eq_true, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, congrFun', List.length and 46 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem OeisA114137.a_4 line 86 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA114137.a, OfNat.ofNat
Proof uses: OeisA114137.a_eq_of, of_eq_true, And, Nat.IsSemiprime, Odd, Eq.trans, True, congr, congrArg, Ne, ArithmeticFunction, ArithmeticFunction.cardFactors, eq_true, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, congrFun', List.length and 59 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
theorem OeisA114137.a_5 line 95 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA114137.a, OfNat.ofNat
Proof uses: OeisA114137.a_eq_of, of_eq_true, And, Nat.IsSemiprime, Odd, Eq.trans, True, congr, congrArg, Ne, ArithmeticFunction, ArithmeticFunction.cardFactors, eq_true, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, congrFun', List.length and 50 more
34 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem OeisA114137.conjecture1 In this powers of 2 sequence, does 1 occur infinitely often? sorry line 106 source
Source documentation
In this powers of 2 sequence, does 1 occur infinitely often?

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, Eq, OeisA114137.a, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture1 :
  answer(sorry) ↔ Set.Infinite {n : ℕ | a n = 1} := by
…
```
Mentions: OeisA114137.a
theorem OeisA114137.conjecture2 Does every odd number occur? sorry line 114 source
Source documentation
Does every odd number occur?

depends on sorry what this means
Statement uses: Iff, True, Nat, Odd, Exists, Eq, OeisA114137.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2 :
  answer(sorry) ↔ ∀ k : ℕ, Odd k → ∃ n : ℕ, a n = k := by
…
```
Mentions: OeisA114137.a