119563.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

7 declarations, 1 contain sorry, 0 declared as axioms

Sum of Fermat number and Mersenne number minus 1: $2^{2^n} + 2^n - 1$

Define $F(n) = 2^{2^n} + 1$ (the $n$-th Fermat number) and $M(n) = 2^n - 1$ (the $n$-th Mersenne number). Then $a(n) = F(n) + M(n) - 1 = 2^{2^n} + 2^n - 1$.

References:

A119563

Imports (1)

FormalConjecturesUtil

def OeisA119563.a $a(n) = 2^{2^n} + 2^n - 1$ line 34 source
Source documentation
$a(n) = 2^{2^n} + 2^n - 1$.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: HSub.hSub, HAdd.hAdd, HPow.hPow, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ := 2 ^ (2 ^ n) + 2 ^ n - 1
…
```
theorem OeisA119563.a_0 Value of the sequence a at 0 line 38 source
Source documentation
Value of the sequence a at 0.

standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA119563.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_0 : a 0 = 2 := by rfl
…
```
Mentions: OeisA119563.a
theorem OeisA119563.a_1 Value of the sequence a at 1 line 42 source
Source documentation
Value of the sequence a at 1.

standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA119563.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_1 : a 1 = 5 := by rfl
…
```
Mentions: OeisA119563.a
theorem OeisA119563.a_2 Value of the sequence a at 2 line 46 source
Source documentation
Value of the sequence a at 2.

standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA119563.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_2 : a 2 = 19 := by rfl
…
```
Mentions: OeisA119563.a
theorem OeisA119563.a_3 Value of the sequence a at 3 line 50 source
Source documentation
Value of the sequence a at 3.

standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA119563.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_3 : a 3 = 263 := by rfl
…
```
Mentions: OeisA119563.a
theorem OeisA119563.a_4 Value of the sequence a at 4 line 54 source
Source documentation
Value of the sequence a at 4.

standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA119563.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_4 : a 4 = 65551 := by rfl
…
```
Mentions: OeisA119563.a
theorem OeisA119563.conjecture The first 5 entries are primes sorry line 59 source
Source documentation
The first 5 entries are primes. Are there infinitely many primes in this sequence?

depends on sorry what this means
Statement uses: Iff, True, Set.Infinite, Nat, Set.ofPred, Nat.Prime, OeisA119563.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 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) ↔ Set.Infinite {n : ℕ | (a n).Prime} := by
…
```
Mentions: OeisA119563.a