157225.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

8 declarations, 1 contain sorry, 0 declared as axioms

Representations as $p + 2^x + 7 \cdot 2^y$ with $p \equiv 5 \pmod 6$

Number of ways to write the $n$-th positive odd integer in the form $p + 2^x + 7 \cdot 2^y$ with $p$ a prime congruent to $5 \bmod 6$ and $x, y$ positive integers. $$a(n) = \left|\left\{(p, x, y) : p + 2^x + 7 \cdot 2^y = 2n - 1 \text{ with } p \text{ prime}, p \equiv 5 \pmod 6, x, y \in \mathbb{Z}^+\right\}\right|.$$

References:

A157225
Z.-W. Sun, "Mixed sums of primes and other terms", arXiv preprint arXiv:0901.3075 [math.NT], 2009.

Imports (1)

FormalConjecturesUtil

def OeisA157225.a Number of representations of $2n - 1$ as $p + 2^x + 7 \cdot 2^y$ with $p \equiv 5 \pmod 6$ line 38 source
Source documentation
Number of representations of $2n - 1$ as $p + 2^x + 7 \cdot 2^y$ with $p \equiv 5 \pmod 6$.

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 OeisA157225.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, OeisA157225.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: OeisA157225.a
theorem OeisA157225.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, OeisA157225.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: OeisA157225.a
theorem OeisA157225.a_11 Value of the sequence a at 11 line 58 source
Source documentation
Value of the sequence a at 11.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA157225.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_11 : a 11 = 1 := by decide
…
```
Mentions: OeisA157225.a
theorem OeisA157225.a_12 Value of the sequence a at 12 line 62 source
Source documentation
Value of the sequence a at 12.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA157225.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_12 : a 12 = 1 := by decide
…
```
Mentions: OeisA157225.a
theorem OeisA157225.a_13 Value of the sequence a at 13 line 66 source
Source documentation
Value of the sequence a at 13.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA157225.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_13 : a 13 = 0 := by decide
…
```
Mentions: OeisA157225.a
theorem OeisA157225.a_14 Value of the sequence a at 14 line 70 source
Source documentation
Value of the sequence a at 14.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA157225.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_14 : a 14 = 2 := by decide
…
```
Mentions: OeisA157225.a
theorem OeisA157225.conjecture sorry line 85 source
depends on sorry what this means
Statement uses: Not, Nat, LT.lt, OfNat.ofNat, Iff, Eq, OeisA157225.a, Or
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.