157237.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

8 declarations, 1 contain sorry, 0 declared as axioms

Representations as $p + 2^x + 11 \cdot 2^y$ with $p \equiv 1 \pmod 6$

Number of ways to write the $n$-th positive odd integer in the form $p + 2^x + 11 \cdot 2^y$ with $p$ a prime congruent to $1 \bmod 6$ and $x, y$ positive integers. $$a(n) = \left|\left\{(p, x, y) : p + 2^x + 11 \cdot 2^y = 2n - 1 \text{ with } p \text{ prime}, p \equiv 1 \pmod 6, x, y \in \mathbb{Z}^+\right\}\right|.$$

References:

A157237
Z.-W. Sun, "Mixed sums of primes and other terms", arXiv preprint arXiv:0901.3075 [math.NT], 2009.

Imports (1)

FormalConjecturesUtil

def OeisA157237.a Number of representations of $2n - 1$ as $p + 2^x + 11 \cdot 2^y$ with $p \equiv 1 \pmod 6$ line 38 source
Source documentation
Number of representations of $2n - 1$ as $p + 2^x + 11 \cdot 2^y$ with $p \equiv 1 \pmod 6$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, HSub.hSub, HMul.hMul, Finset.sum, Finset.Icc, And, LT.lt, HAdd.hAdd, HPow.hPow, Nat.Prime, HMod.hMod
21 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA157237.a_1 Value of the sequence a at 1 line 50 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA157237.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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_1 : a 1 = 0 := by decide
…
```
Mentions: OeisA157237.a
theorem OeisA157237.a_2 Value of the sequence a at 2 line 54 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA157237.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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_2 : a 2 = 0 := by decide
…
```
Mentions: OeisA157237.a
theorem OeisA157237.a_16 Value of the sequence a at 16 line 58 source
Source documentation
Value of the sequence a at 16.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA157237.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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_16 : a 16 = 1 := by decide
…
```
Mentions: OeisA157237.a
theorem OeisA157237.a_17 Value of the sequence a at 17 line 62 source
Source documentation
Value of the sequence a at 17.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA157237.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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_17 : a 17 = 1 := by decide
…
```
Mentions: OeisA157237.a
theorem OeisA157237.a_18 Value of the sequence a at 18 line 66 source
Source documentation
Value of the sequence a at 18.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA157237.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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_18 : a 18 = 0 := by decide
…
```
Mentions: OeisA157237.a
theorem OeisA157237.a_19 Value of the sequence a at 19 line 70 source
Source documentation
Value of the sequence a at 19.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA157237.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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_19 : a 19 = 2 := by decide
…
```
Mentions: OeisA157237.a
theorem OeisA157237.conjecture On Feb sorry line 79 source
Source documentation
On Feb. 24, 2009, Zhi-Wei Sun conjectured that $a(n) = 0$ if and only if $n < 16$ or $n \in \{18, 21, 24, 51, 84, 1011, 59586\}$; in other words, except for $35, 41, 47, 101, 167, 2021, 119171$, any odd integer greater than $30$ can be written as the sum of a prime congruent to $1 \bmod 6$, a positive power of $2$ and eleven times a positive power of $2$.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Iff, Eq, OeisA157237.a, Or, LE.le
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) (hn : 0 < n) :
    a n = 0 ↔ n ≤ 15 ∨ n = 18 ∨ n = 21 ∨ n = 24 ∨ n = 51 ∨ n = 84 ∨ n = 1011 ∨ n = 59586 := by
…
```
Mentions: OeisA157237.a