FormalConjectures.OEIS.«228828»

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import FormalConjecturesUtil

Numbers $n$ such that $n^2 + \pi(n)$ is prime.

Reference: A228828


namespace OeisA228828


open scoped Nat.Prime

Numbers n such that $n^2 + \pi(n)$ is prime.

noncomputable def a (n : ℕ) : ℕ := n.nth (fun n => (n ^ 2 + π n).Prime)


@[category test, AMS 11]
theorem a_0 : a 0 = 2 := by⊢ a 0 = 2
  unfold a⊢ Nat.nth (fun n => Nat.Prime (n ^ 2 + π n)) 0 = 2
  convert Nat.nth_count _h.e'_2.h.e'_2⊢ 0 = Nat.count (fun n => Nat.Prime (n ^ 2 + π n)) 2convert_2⊢ DecidablePred fun n => Nat.Prime (n ^ 2 + π n)convert_4⊢ Nat.Prime (2 ^ 2 + π 2)
  ·h.e'_2.h.e'_2⊢ 0 = Nat.count (fun n => Nat.Prime (n ^ 2 + π n)) 2 norm_num [Nat.count_succ]All goals completed! 🐙
  ·convert_2⊢ DecidablePred fun n => Nat.Prime (n ^ 2 + π n) exact Classical.decPred fun n ↦ Nat.Prime (n ^ 2 + π n)All goals completed! 🐙
  ·convert_4⊢ Nat.Prime (2 ^ 2 + π 2) decideAll goals completed! 🐙


@[category test, AMS 11]
theorem a_1 : a 1 = 3 := by⊢ a 1 = 3
  unfold a⊢ Nat.nth (fun n => Nat.Prime (n ^ 2 + π n)) 1 = 3
  convert Nat.nth_count _h.e'_2.h.e'_2⊢ 1 = Nat.count (fun n => Nat.Prime (n ^ 2 + π n)) 3convert_2⊢ DecidablePred fun n => Nat.Prime (n ^ 2 + π n)convert_4⊢ Nat.Prime (3 ^ 2 + π 3)
  ·h.e'_2.h.e'_2⊢ 1 = Nat.count (fun n => Nat.Prime (n ^ 2 + π n)) 3 norm_num [Nat.count_succ]h.e'_2.h.e'_2⊢ Nat.Prime (4 + π 2)
    decideAll goals completed! 🐙
  ·convert_2⊢ DecidablePred fun n => Nat.Prime (n ^ 2 + π n) exact Classical.decPred fun n ↦ Nat.Prime (n ^ 2 + π n)All goals completed! 🐙
  ·convert_4⊢ Nat.Prime (3 ^ 2 + π 3) decideAll goals completed! 🐙


@[category test, AMS 11]
theorem a_2 : a 2 = 7 := by⊢ a 2 = 7
  unfold a⊢ Nat.nth (fun n => Nat.Prime (n ^ 2 + π n)) 2 = 7
  convert Nat.nth_count _h.e'_2.h.e'_2⊢ 2 = Nat.count (fun n => Nat.Prime (n ^ 2 + π n)) 7convert_2⊢ DecidablePred fun n => Nat.Prime (n ^ 2 + π n)convert_4⊢ Nat.Prime (7 ^ 2 + π 7)
  ·h.e'_2.h.e'_2⊢ 2 = Nat.count (fun n => Nat.Prime (n ^ 2 + π n)) 7 norm_num [Nat.count_succ]h.e'_2.h.e'_2⊢ 2 =
  ((((if Nat.Prime (4 + π 2) then 1 else 0) + if Nat.Prime (9 + π 3) then 1 else 0) +
        if Nat.Prime (16 + π 4) then 1 else 0) +
      if Nat.Prime (25 + π 5) then 1 else 0) +
    if Nat.Prime (36 + π 6) then 1 else 0
    rflAll goals completed! 🐙
  ·convert_2⊢ DecidablePred fun n => Nat.Prime (n ^ 2 + π n) exact Classical.decPred fun n ↦ Nat.Prime (n ^ 2 + π n)All goals completed! 🐙
  ·convert_4⊢ Nat.Prime (7 ^ 2 + π 7) decideAll goals completed! 🐙

Conjecture: the sequence A228828 is infinite.

@[category research open, AMS 11]
theorem declaration uses 'sorry'a.infinite : {a n | n}.Infinite := by⊢ {x | ∃ n, a n = x}.Infinite
  sorryAll goals completed! 🐙


end OeisA228828