Odious primes minus evil primes among first $n$ primes

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import FormalConjecturesUtil

/-!
# 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](https://oeis.org/A130911)-/

namespace OeisA130911

/-- Parity sign of binary weight: $+1$ if popcount is odd, $-1$ if even. -/
def signWeight (k : ℕ) : ℤ :=
  if (Nat.digits 2 k).sum % 2 = 1 then 1 else -1

/-- $a(n) = \sum_{i=0}^{n-1} \mathrm{signWeight}(p_i)$ where $p_i$ is the $i$-th prime. -/
noncomputable def a (n : ℕ) : ℤ :=
  ∑ i ∈ Finset.range n, signWeight (Nat.nth Nat.Prime i)

/-- Value of the sequence `a` at 0. -/
@[category test, AMS 11]
theorem a_0 : a 0 = 0 := by rfl

/-- Value of the sequence `a` at 1. -/
@[category test, AMS 11]
theorem a_1 : a 1 = 1 := by
  unfold a
  have h : Finset.range 1 = {0} := by decide
  rw [h, Finset.sum_singleton]
  have h0 : Nat.nth Nat.Prime 0 = 2 := Nat.nth_prime_zero_eq_two
  rw [h0]
  decide +native

/-- Value of the sequence `a` at 2. -/
@[category test, AMS 11]
theorem a_2 : a 2 = 0 := by
  unfold a
  have h : Finset.range 2 = {0, 1} := by decide
  rw [h, Finset.sum_pair (by decide)]
  have h0 : Nat.nth Nat.Prime 0 = 2 := Nat.nth_prime_zero_eq_two
  have h1 : Nat.nth Nat.Prime 1 = 3 := Nat.nth_prime_one_eq_three
  rw [h0, h1]
  decide +native

/-- Value of the sequence `a` at 3. -/
@[category test, AMS 11]
theorem a_3 : a 3 = -1 := by
  unfold a
  have h : Finset.range 3 = {0, 1, 2} := by decide
  rw [h]
  have h0 : Nat.nth Nat.Prime 0 = 2 := Nat.nth_prime_zero_eq_two
  have h1 : Nat.nth Nat.Prime 1 = 3 := Nat.nth_prime_one_eq_three
  have h2 : Nat.nth Nat.Prime 2 = 5 := Nat.nth_prime_two_eq_five
  rw [Finset.sum_insert (by decide), Finset.sum_pair (by decide)]
  rw [h0, h1, h2]
  decide +native

/-- Value of the sequence `a` at 4. -/
@[category test, AMS 11]
theorem a_4 : a 4 = 0 := by
  unfold a
  have h : Finset.range 4 = {0, 1, 2, 3} := by decide
  rw [h]
  have h0 : Nat.nth Nat.Prime 0 = 2 := Nat.nth_prime_zero_eq_two
  have h1 : Nat.nth Nat.Prime 1 = 3 := Nat.nth_prime_one_eq_three
  have h2 : Nat.nth Nat.Prime 2 = 5 := Nat.nth_prime_two_eq_five
  have h3 : Nat.nth Nat.Prime 3 = 7 := Nat.nth_prime_three_eq_seven
  rw [Finset.sum_insert (by decide), Finset.sum_insert (by decide),
    Finset.sum_pair (by decide)]
  rw [h0, h1, h2, h3]
  decide +native

/--
Shevelev conjectures that $a(n) \ge 0$ for $n > 3$.-/
@[category research open, AMS 11]
theorem conjecture (n : ℕ) (hn : 3 < n) : a n ≥ 0 := by
  sorry

end OeisA130911

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