FormalConjectures.OEIS.«109227»

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import FormalConjecturesUtil

Conjectures associated with A109227

Binary strings that have 1's where primes occur, 0's elsewhere and every term ends with the $n$-th prime index.

Conjecture: $a(2)$ and $a(121)$ are primes. Are there any more?

References:

A109227

namespace OeisA109227open Nat List

The primary defining sequence a. $a(n)$ is the natural number whose decimal digits are given by the binary string of length $p_n$ where the $k$-th digit is $1$ if $k$ is prime and $0$ otherwise, with leading zeros dropped.

noncomputable def a (n : ℕ) : ℕ :=
  if n = 0 then 0 else
  let pN : ℕ := Nat.nth Nat.Prime (n - 1)
  let primeBitsFull : List ℕ := (range (pN + 1)).map (fun i =>
    if i.Prime then 1 else 0
  )
  let primeBitsTrimmed := primeBitsFull.dropWhile (· = 0)
  ofDigits 10 primeBitsTrimmed.reverse

Term theorems verifying the first few values of the sequence against the official OEIS b-file

@[category test, AMS 11]
theorem a_0 : a 0 = 0 := by⊢ a 0 = 0
  rflAll goals completed! 🐙



@[category test, AMS 11]
theorem a_1 : a 1 = 1 := by⊢ a 1 = 1
  dsimp [a]⊢ ofDigits 10
    (dropWhile (fun x ↦ decide (x = 0))
        (map (fun i ↦ if Nat.Prime i then 1 else 0) (range (nth Nat.Prime 0 + 1)))).reverse =
  1
  rw [Nat.nth_prime_zero_eq_two⊢ ofDigits 10 (dropWhile (fun x ↦ decide (x = 0)) (map (fun i ↦ if Nat.Prime i then 1 else 0) (range (2 + 1)))).reverse =
  1⊢ ofDigits 10 (dropWhile (fun x ↦ decide (x = 0)) (map (fun i ↦ if Nat.Prime i then 1 else 0) (range (2 + 1)))).reverse =
  1]⊢ ofDigits 10 (dropWhile (fun x ↦ decide (x = 0)) (map (fun i ↦ if Nat.Prime i then 1 else 0) (range (2 + 1)))).reverse =
  1
  decideAll goals completed! 🐙



@[category test, AMS 11]
theorem a_2 : a 2 = 11 := by⊢ a 2 = 11
  dsimp [a]⊢ ofDigits 10
    (dropWhile (fun x ↦ decide (x = 0))
        (map (fun i ↦ if Nat.Prime i then 1 else 0) (range (nth Nat.Prime 1 + 1)))).reverse =
  11
  rw [Nat.nth_prime_one_eq_three⊢ ofDigits 10 (dropWhile (fun x ↦ decide (x = 0)) (map (fun i ↦ if Nat.Prime i then 1 else 0) (range (3 + 1)))).reverse =
  11⊢ ofDigits 10 (dropWhile (fun x ↦ decide (x = 0)) (map (fun i ↦ if Nat.Prime i then 1 else 0) (range (3 + 1)))).reverse =
  11]⊢ ofDigits 10 (dropWhile (fun x ↦ decide (x = 0)) (map (fun i ↦ if Nat.Prime i then 1 else 0) (range (3 + 1)))).reverse =
  11
  decideAll goals completed! 🐙



@[category test, AMS 11]
theorem a_3 : a 3 = 1101 := by⊢ a 3 = 1101
  dsimp [a]⊢ ofDigits 10
    (dropWhile (fun x ↦ decide (x = 0))
        (map (fun i ↦ if Nat.Prime i then 1 else 0) (range (nth Nat.Prime 2 + 1)))).reverse =
  1101
  rw [Nat.nth_prime_two_eq_five⊢ ofDigits 10 (dropWhile (fun x ↦ decide (x = 0)) (map (fun i ↦ if Nat.Prime i then 1 else 0) (range (5 + 1)))).reverse =
  1101⊢ ofDigits 10 (dropWhile (fun x ↦ decide (x = 0)) (map (fun i ↦ if Nat.Prime i then 1 else 0) (range (5 + 1)))).reverse =
  1101]⊢ ofDigits 10 (dropWhile (fun x ↦ decide (x = 0)) (map (fun i ↦ if Nat.Prime i then 1 else 0) (range (5 + 1)))).reverse =
  1101
  decideAll goals completed! 🐙



@[category test, AMS 11]
theorem a_4 : a 4 = 110101 := by⊢ a 4 = 110101
  dsimp [a]⊢ ofDigits 10
    (dropWhile (fun x ↦ decide (x = 0))
        (map (fun i ↦ if Nat.Prime i then 1 else 0) (range (nth Nat.Prime 3 + 1)))).reverse =
  110101
  rw [Nat.nth_prime_three_eq_seven⊢ ofDigits 10 (dropWhile (fun x ↦ decide (x = 0)) (map (fun i ↦ if Nat.Prime i then 1 else 0) (range (7 + 1)))).reverse =
  110101⊢ ofDigits 10 (dropWhile (fun x ↦ decide (x = 0)) (map (fun i ↦ if Nat.Prime i then 1 else 0) (range (7 + 1)))).reverse =
  110101]⊢ ofDigits 10 (dropWhile (fun x ↦ decide (x = 0)) (map (fun i ↦ if Nat.Prime i then 1 else 0) (range (7 + 1)))).reverse =
  110101
  decideAll goals completed! 🐙

Conjecture: $a(2)$ and $a(121)$ are primes. Are there any more?

@[category research open, AMS 11]
theorem conjecture :
    answer(sorry) ↔ ∃ n > 0, n ≠ 2 ∧ n ≠ 121 ∧ (a n).Prime := by⊢ True ↔ ∃ n > 0, n ≠ 2 ∧ n ≠ 121 ∧ Nat.Prime (a n)
  sorryAll goals completed! 🐙

end OeisA109227