113271.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

Ascending descending base exponent transform of $2^n$

References:

A113271

Imports (1)

FormalConjecturesUtil

def OeisA113271.a The primary defining sequence a line 36 source
Source documentation
The primary defining sequence a. $a(n)$ is the ascending descending base exponent transform of $2^n$. $$a(n) = \sum_{i=0}^n 2^{i \cdot 2^{n-i}}$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HPow.hPow, HMul.hMul, HSub.hSub
12 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA113271.a_0 line 40 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA113271.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 OeisA113271.a_1 line 43 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA113271.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 OeisA113271.a_2 line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA113271.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 OeisA113271.a_3 line 49 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA113271.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 OeisA113271.a_4 line 52 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA113271.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 OeisA113271.a_4_prime line 55 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat.Prime, OeisA113271.a, OfNat.ofNat, Nat
Proof uses: Eq.mpr, id, Eq, congrArg, OeisA113271.a_4, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_4, Mathlib.Meta.NormNum.IsNat.raw_refl, Mathlib.Meta.NormNum.minFacHelper_3, Bool.false, Mathlib.Meta.NormNum.minFacHelper_2, Mathlib.Meta.NormNum.not_prime_mul_of_ble, Mathlib.Meta.NormNum.minFacHelper_0
3 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA113271.a_5 The next term from the defining formula and the official OEIS b-file line 61 source
Source documentation
The next term from the defining formula and the official OEIS b-file.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA113271.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_5 : a 5 = 135457 := by rfl
…
```
Mentions: OeisA113271.a
theorem OeisA113271.a_5_not_prime The term a(5) = 135457 = 7 * 37 * 523 is composite line 65 source
Source documentation
The term a(5) = 135457 = 7 * 37 * 523 is composite.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, Nat.Prime, OeisA113271.a, OfNat.ofNat, Nat
Proof uses: Eq.mpr, id, Eq, congrArg, OeisA113271.a_5, Mathlib.Meta.NormNum.isNat_not_prime, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.not_prime_mul_of_ble, Bool, Bool.false
3 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem a_5_not_prime : ¬ (a 5).Prime := by
…
```
Mentions: OeisA113271.a
theorem OeisA113271.conjecture1 The first prime terms in this (always odd) sequence are $a(1) = 3$, $a(3) = 41$, and $a(4) = 593$ sorry line 78 source
Source documentation
The first prime terms in this (always odd) sequence are $a(1) = 3$, $a(3) = 41$, and $a(4) = 593$. What is the next prime?
The OEIS comment currently says $a(5) = 543$ , but this conflicts with its defining formula, b-file, and examples: the actual index-five term is the composite number 135457. Consequently the search for the next prime must begin after index 4, not after index 5.

depends on sorry what this means
Statement uses: Eq, Nat, sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, OeisA113271.a, InfSet.sInf, Set.ofPred, And, LT.lt, Nat.Prime
6 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) = a (sInf {n : ℕ | 4 < n ∧ (a n).Prime}) := by
…
```
Mentions: OeisA113271.a