130911.lean`

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

8 declarations, 1 contain sorry, 0 declared as axioms

Odious primes minus evil primes among first $n$ primes

$a(n)$ is the number of primes with odd binary weight (odious primes) among the first $n$ primes minus the number with even binary weight (evil primes).

References:

A130911

Imports (1)

FormalConjecturesUtil

def OeisA130911.signWeight Parity sign of binary weight: $+1$ if popcount is odd, $-1$ if even line 35 source
Source documentation
Parity sign of binary weight: $+1$ if popcount is odd, $-1$ if even.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: ite, Eq, HMod.hMod, List.sum, Nat.digits, OfNat.ofNat, Neg.neg
9 instances, coercions and auxiliary constants left out.
```
def signWeight (k : ℕ) : ℤ :=
…
```
def OeisA130911.a $a(n) = \sum_{i=0}^{n-1} \mathrm{signWeight}(p_i)$ where $p_i$ is the $i$-th prime line 39 source
Source documentation
$a(n) = \sum_{i=0}^{n-1} \mathrm{signWeight}(p_i)$ where $p_i$ is the $i$-th prime.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: Finset.sum, Finset.range, OeisA130911.signWeight, Nat.nth, Nat.Prime
1 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℤ :=
…
```
theorem OeisA130911.a_0 Value of the sequence a at 0 line 44 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA130911.a, OfNat.ofNat, Nat
Proof uses: 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_0 : a 0 = 0 := by rfl
…
```
Mentions: OeisA130911.a
theorem OeisA130911.a_1 Value of the sequence a at 1 line 48 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA130911.a, OfNat.ofNat, Nat
Proof uses: id, Finset, Finset.range, Singleton.singleton, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, Finset.sum, OeisA130911.signWeight, Nat.nth, Nat.Prime, congrArg, Finset.sum_singleton, Nat.nth_prime_zero_eq_two
9 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 1 := by
…
```
Mentions: OeisA130911.a
theorem OeisA130911.a_2 Value of the sequence a at 2 line 58 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA130911.a, OfNat.ofNat, Nat
Proof uses: id, Finset, Finset.range, Insert.insert, Singleton.singleton, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, Finset.sum, OeisA130911.signWeight, Nat.nth, Nat.Prime, congrArg, HAdd.hAdd, Finset.sum_pair, Ne, Nat.nth_prime_zero_eq_two and 1 more
16 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 0 := by
…
```
Mentions: OeisA130911.a
theorem OeisA130911.a_3 Value of the sequence a at 3 line 69 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA130911.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: id, Finset, Finset.range, Insert.insert, Singleton.singleton, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, Finset.sum, OeisA130911.signWeight, Nat.nth, Nat.Prime, congrArg, Nat.nth_prime_zero_eq_two, Nat.nth_prime_one_eq_three, Nat.nth_prime_two_eq_five, HAdd.hAdd and 5 more
21 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = -1 := by
…
```
Mentions: OeisA130911.a
theorem OeisA130911.a_4 Value of the sequence a at 4 line 82 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA130911.a, OfNat.ofNat, Nat
Proof uses: id, Finset, Finset.range, Insert.insert, Singleton.singleton, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, Finset.sum, OeisA130911.signWeight, Nat.nth, Nat.Prime, congrArg, Nat.nth_prime_zero_eq_two, Nat.nth_prime_one_eq_three, Nat.nth_prime_two_eq_five, Nat.nth_prime_three_eq_seven and 6 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 0 := by
…
```
Mentions: OeisA130911.a
theorem OeisA130911.conjecture Shevelev conjectures that $a(n) \ge 0$ for $n > 3$ sorry line 98 source
Source documentation
Shevelev conjectures that $a(n) \ge 0$ for $n > 3$.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, GE.ge, Int, OeisA130911.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 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 : 3 < n) : a n ≥ 0 := by
…
```
Mentions: OeisA130911.a