FormalConjectures.OEIS.«130911»

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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

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⊢ a 0 = 0 rflAll goals completed! 🐙

Value of the sequence a at 1.


@[category test, AMS 11]
theorem a_1 : a 1 = 1 := by⊢ a 1 = 1
  unfold a⊢ ∑ i ∈ Finset.range 1, signWeight (Nat.nth Nat.Prime i) = 1
  have h : Finset.range 1 = {0} := by⊢ a 1 = 1h:Finset.range 1 = {0}⊢ ∑ i ∈ Finset.range 1, signWeight (Nat.nth Nat.Prime i) = 1 decideh:Finset.range 1 = {0}⊢ ∑ i ∈ Finset.range 1, signWeight (Nat.nth Nat.Prime i) = 1h:Finset.range 1 = {0}⊢ ∑ i ∈ Finset.range 1, signWeight (Nat.nth Nat.Prime i) = 1
  rw [h,h:Finset.range 1 = {0}⊢ ∑ i ∈ {0}, signWeight (Nat.nth Nat.Prime i) = 1h:Finset.range 1 = {0}⊢ signWeight (Nat.nth Nat.Prime 0) = 1 Finset.sum_singletonh:Finset.range 1 = {0}⊢ signWeight (Nat.nth Nat.Prime 0) = 1h:Finset.range 1 = {0}⊢ signWeight (Nat.nth Nat.Prime 0) = 1]h:Finset.range 1 = {0}⊢ signWeight (Nat.nth Nat.Prime 0) = 1
  have h0 : Nat.nth Nat.Prime 0 = 2 := Nat.nth_prime_zero_eq_twoh:Finset.range 1 = {0}h0:Nat.nth Nat.Prime 0 = 2⊢ signWeight (Nat.nth Nat.Prime 0) = 1
  rw [h0h:Finset.range 1 = {0}h0:Nat.nth Nat.Prime 0 = 2⊢ signWeight 2 = 1h:Finset.range 1 = {0}h0:Nat.nth Nat.Prime 0 = 2⊢ signWeight 2 = 1]h:Finset.range 1 = {0}h0:Nat.nth Nat.Prime 0 = 2⊢ signWeight 2 = 1
  decide +nativeAll goals completed! 🐙

Value of the sequence a at 2.


@[category test, AMS 11]
theorem a_2 : a 2 = 0 := by⊢ a 2 = 0
  unfold a⊢ ∑ i ∈ Finset.range 2, signWeight (Nat.nth Nat.Prime i) = 0
  have h : Finset.range 2 = {0, 1} := by⊢ a 2 = 0h:Finset.range 2 = {0, 1}⊢ ∑ i ∈ Finset.range 2, signWeight (Nat.nth Nat.Prime i) = 0 decideh:Finset.range 2 = {0, 1}⊢ ∑ i ∈ Finset.range 2, signWeight (Nat.nth Nat.Prime i) = 0h:Finset.range 2 = {0, 1}⊢ ∑ i ∈ Finset.range 2, signWeight (Nat.nth Nat.Prime i) = 0
  rw [h,h:Finset.range 2 = {0, 1}⊢ ∑ i ∈ {0, 1}, signWeight (Nat.nth Nat.Prime i) = 0h:Finset.range 2 = {0, 1}⊢ signWeight (Nat.nth Nat.Prime 0) + signWeight (Nat.nth Nat.Prime 1) = 0 Finset.sum_pair (byh:Finset.range 2 = {0, 1}⊢ 0 ≠ 1h:Finset.range 2 = {0, 1}⊢ signWeight (Nat.nth Nat.Prime 0) + signWeight (Nat.nth Nat.Prime 1) = 0 decideAll goals completed! 🐙h:Finset.range 2 = {0, 1}⊢ signWeight (Nat.nth Nat.Prime 0) + signWeight (Nat.nth Nat.Prime 1) = 0)]h:Finset.range 2 = {0, 1}⊢ signWeight (Nat.nth Nat.Prime 0) + signWeight (Nat.nth Nat.Prime 1) = 0
  have h0 : Nat.nth Nat.Prime 0 = 2 := Nat.nth_prime_zero_eq_twoh:Finset.range 2 = {0, 1}h0:Nat.nth Nat.Prime 0 = 2⊢ signWeight (Nat.nth Nat.Prime 0) + signWeight (Nat.nth Nat.Prime 1) = 0
  have h1 : Nat.nth Nat.Prime 1 = 3 := Nat.nth_prime_one_eq_threeh:Finset.range 2 = {0, 1}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3⊢ signWeight (Nat.nth Nat.Prime 0) + signWeight (Nat.nth Nat.Prime 1) = 0
  rw [h0,h:Finset.range 2 = {0, 1}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3⊢ signWeight 2 + signWeight (Nat.nth Nat.Prime 1) = 0h:Finset.range 2 = {0, 1}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3⊢ signWeight 2 + signWeight 3 = 0 h1h:Finset.range 2 = {0, 1}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3⊢ signWeight 2 + signWeight 3 = 0h:Finset.range 2 = {0, 1}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3⊢ signWeight 2 + signWeight 3 = 0]h:Finset.range 2 = {0, 1}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3⊢ signWeight 2 + signWeight 3 = 0
  decide +nativeAll goals completed! 🐙

Value of the sequence a at 3.


@[category test, AMS 11]
theorem a_3 : a 3 = -1 := by⊢ a 3 = -1
  unfold a⊢ ∑ i ∈ Finset.range 3, signWeight (Nat.nth Nat.Prime i) = -1
  have h : Finset.range 3 = {0, 1, 2} := by⊢ a 3 = -1h:Finset.range 3 = {0, 1, 2}⊢ ∑ i ∈ Finset.range 3, signWeight (Nat.nth Nat.Prime i) = -1 decideh:Finset.range 3 = {0, 1, 2}⊢ ∑ i ∈ Finset.range 3, signWeight (Nat.nth Nat.Prime i) = -1h:Finset.range 3 = {0, 1, 2}⊢ ∑ i ∈ Finset.range 3, signWeight (Nat.nth Nat.Prime i) = -1
  rw [hh:Finset.range 3 = {0, 1, 2}⊢ ∑ i ∈ {0, 1, 2}, signWeight (Nat.nth Nat.Prime i) = -1h:Finset.range 3 = {0, 1, 2}⊢ ∑ i ∈ {0, 1, 2}, signWeight (Nat.nth Nat.Prime i) = -1]h:Finset.range 3 = {0, 1, 2}⊢ ∑ i ∈ {0, 1, 2}, signWeight (Nat.nth Nat.Prime i) = -1
  have h0 : Nat.nth Nat.Prime 0 = 2 := Nat.nth_prime_zero_eq_twoh:Finset.range 3 = {0, 1, 2}h0:Nat.nth Nat.Prime 0 = 2⊢ ∑ i ∈ {0, 1, 2}, signWeight (Nat.nth Nat.Prime i) = -1
  have h1 : Nat.nth Nat.Prime 1 = 3 := Nat.nth_prime_one_eq_threeh:Finset.range 3 = {0, 1, 2}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3⊢ ∑ i ∈ {0, 1, 2}, signWeight (Nat.nth Nat.Prime i) = -1
  have h2 : Nat.nth Nat.Prime 2 = 5 := Nat.nth_prime_two_eq_fiveh:Finset.range 3 = {0, 1, 2}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5⊢ ∑ i ∈ {0, 1, 2}, signWeight (Nat.nth Nat.Prime i) = -1
  rw [Finset.sum_insert (byh:Finset.range 3 = {0, 1, 2}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5⊢ 0 ∉ {1, 2}h:Finset.range 3 = {0, 1, 2}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5⊢ signWeight (Nat.nth Nat.Prime 0) + (signWeight (Nat.nth Nat.Prime 1) + signWeight (Nat.nth Nat.Prime 2)) = -1 decideAll goals completed! 🐙h:Finset.range 3 = {0, 1, 2}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5⊢ signWeight (Nat.nth Nat.Prime 0) + (signWeight (Nat.nth Nat.Prime 1) + signWeight (Nat.nth Nat.Prime 2)) = -1), Finset.sum_pair (byh:Finset.range 3 = {0, 1, 2}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5⊢ 1 ≠ 2h:Finset.range 3 = {0, 1, 2}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5⊢ signWeight (Nat.nth Nat.Prime 0) + (signWeight (Nat.nth Nat.Prime 1) + signWeight (Nat.nth Nat.Prime 2)) = -1 decideAll goals completed! 🐙h:Finset.range 3 = {0, 1, 2}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5⊢ signWeight (Nat.nth Nat.Prime 0) + (signWeight (Nat.nth Nat.Prime 1) + signWeight (Nat.nth Nat.Prime 2)) = -1)]h:Finset.range 3 = {0, 1, 2}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5⊢ signWeight (Nat.nth Nat.Prime 0) + (signWeight (Nat.nth Nat.Prime 1) + signWeight (Nat.nth Nat.Prime 2)) = -1
  rw [h0,h:Finset.range 3 = {0, 1, 2}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5⊢ signWeight 2 + (signWeight (Nat.nth Nat.Prime 1) + signWeight (Nat.nth Nat.Prime 2)) = -1h:Finset.range 3 = {0, 1, 2}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5⊢ signWeight 2 + (signWeight 3 + signWeight 5) = -1 h1,h:Finset.range 3 = {0, 1, 2}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5⊢ signWeight 2 + (signWeight 3 + signWeight (Nat.nth Nat.Prime 2)) = -1h:Finset.range 3 = {0, 1, 2}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5⊢ signWeight 2 + (signWeight 3 + signWeight 5) = -1 h2h:Finset.range 3 = {0, 1, 2}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5⊢ signWeight 2 + (signWeight 3 + signWeight 5) = -1h:Finset.range 3 = {0, 1, 2}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5⊢ signWeight 2 + (signWeight 3 + signWeight 5) = -1]h:Finset.range 3 = {0, 1, 2}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5⊢ signWeight 2 + (signWeight 3 + signWeight 5) = -1
  decide +nativeAll goals completed! 🐙

Value of the sequence a at 4.


@[category test, AMS 11]
theorem a_4 : a 4 = 0 := by⊢ a 4 = 0
  unfold a⊢ ∑ i ∈ Finset.range 4, signWeight (Nat.nth Nat.Prime i) = 0
  have h : Finset.range 4 = {0, 1, 2, 3} := by⊢ a 4 = 0h:Finset.range 4 = {0, 1, 2, 3}⊢ ∑ i ∈ Finset.range 4, signWeight (Nat.nth Nat.Prime i) = 0 decideh:Finset.range 4 = {0, 1, 2, 3}⊢ ∑ i ∈ Finset.range 4, signWeight (Nat.nth Nat.Prime i) = 0h:Finset.range 4 = {0, 1, 2, 3}⊢ ∑ i ∈ Finset.range 4, signWeight (Nat.nth Nat.Prime i) = 0
  rw [hh:Finset.range 4 = {0, 1, 2, 3}⊢ ∑ i ∈ {0, 1, 2, 3}, signWeight (Nat.nth Nat.Prime i) = 0h:Finset.range 4 = {0, 1, 2, 3}⊢ ∑ i ∈ {0, 1, 2, 3}, signWeight (Nat.nth Nat.Prime i) = 0]h:Finset.range 4 = {0, 1, 2, 3}⊢ ∑ i ∈ {0, 1, 2, 3}, signWeight (Nat.nth Nat.Prime i) = 0
  have h0 : Nat.nth Nat.Prime 0 = 2 := Nat.nth_prime_zero_eq_twoh:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2⊢ ∑ i ∈ {0, 1, 2, 3}, signWeight (Nat.nth Nat.Prime i) = 0
  have h1 : Nat.nth Nat.Prime 1 = 3 := Nat.nth_prime_one_eq_threeh:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3⊢ ∑ i ∈ {0, 1, 2, 3}, signWeight (Nat.nth Nat.Prime i) = 0
  have h2 : Nat.nth Nat.Prime 2 = 5 := Nat.nth_prime_two_eq_fiveh:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5⊢ ∑ i ∈ {0, 1, 2, 3}, signWeight (Nat.nth Nat.Prime i) = 0
  have h3 : Nat.nth Nat.Prime 3 = 7 := Nat.nth_prime_three_eq_sevenh:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ ∑ i ∈ {0, 1, 2, 3}, signWeight (Nat.nth Nat.Prime i) = 0
  rw [Finset.sum_insert (byh:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ 0 ∉ {1, 2, 3}h:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ signWeight (Nat.nth Nat.Prime 0) +
    (signWeight (Nat.nth Nat.Prime 1) + (signWeight (Nat.nth Nat.Prime 2) + signWeight (Nat.nth Nat.Prime 3))) =
  0 decideAll goals completed! 🐙h:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ signWeight (Nat.nth Nat.Prime 0) +
    (signWeight (Nat.nth Nat.Prime 1) + (signWeight (Nat.nth Nat.Prime 2) + signWeight (Nat.nth Nat.Prime 3))) =
  0), Finset.sum_insert (byh:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ 1 ∉ {2, 3}h:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ signWeight (Nat.nth Nat.Prime 0) +
    (signWeight (Nat.nth Nat.Prime 1) + (signWeight (Nat.nth Nat.Prime 2) + signWeight (Nat.nth Nat.Prime 3))) =
  0 decideAll goals completed! 🐙h:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ signWeight (Nat.nth Nat.Prime 0) +
    (signWeight (Nat.nth Nat.Prime 1) + (signWeight (Nat.nth Nat.Prime 2) + signWeight (Nat.nth Nat.Prime 3))) =
  0),
    Finset.sum_pair (byh:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ 2 ≠ 3h:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ signWeight (Nat.nth Nat.Prime 0) +
    (signWeight (Nat.nth Nat.Prime 1) + (signWeight (Nat.nth Nat.Prime 2) + signWeight (Nat.nth Nat.Prime 3))) =
  0 decideAll goals completed! 🐙h:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ signWeight (Nat.nth Nat.Prime 0) +
    (signWeight (Nat.nth Nat.Prime 1) + (signWeight (Nat.nth Nat.Prime 2) + signWeight (Nat.nth Nat.Prime 3))) =
  0)]h:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ signWeight (Nat.nth Nat.Prime 0) +
    (signWeight (Nat.nth Nat.Prime 1) + (signWeight (Nat.nth Nat.Prime 2) + signWeight (Nat.nth Nat.Prime 3))) =
  0
  rw [h0,h:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ signWeight 2 +
    (signWeight (Nat.nth Nat.Prime 1) + (signWeight (Nat.nth Nat.Prime 2) + signWeight (Nat.nth Nat.Prime 3))) =
  0h:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ signWeight 2 + (signWeight 3 + (signWeight 5 + signWeight 7)) = 0 h1,h:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ signWeight 2 + (signWeight 3 + (signWeight (Nat.nth Nat.Prime 2) + signWeight (Nat.nth Nat.Prime 3))) = 0h:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ signWeight 2 + (signWeight 3 + (signWeight 5 + signWeight 7)) = 0 h2,h:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ signWeight 2 + (signWeight 3 + (signWeight 5 + signWeight (Nat.nth Nat.Prime 3))) = 0h:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ signWeight 2 + (signWeight 3 + (signWeight 5 + signWeight 7)) = 0 h3h:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ signWeight 2 + (signWeight 3 + (signWeight 5 + signWeight 7)) = 0h:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ signWeight 2 + (signWeight 3 + (signWeight 5 + signWeight 7)) = 0]h:Finset.range 4 = {0, 1, 2, 3}h0:Nat.nth Nat.Prime 0 = 2h1:Nat.nth Nat.Prime 1 = 3h2:Nat.nth Nat.Prime 2 = 5h3:Nat.nth Nat.Prime 3 = 7⊢ signWeight 2 + (signWeight 3 + (signWeight 5 + signWeight 7)) = 0
  decide +nativeAll goals completed! 🐙

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 := byn:ℕhn:3 < n⊢ a n ≥ 0
  sorryAll goals completed! 🐙

end OeisA130911