FormalConjectures.OEIS.«40»

/-
Copyright 2026 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    https://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/

import FormalConjecturesUtil

The prime numbers

The $n$-th prime number $p_n$, where $p_1 = 2$.

References:

A000040

namespace OeisA40

The sequence of prime numbers, with $a(0) = 0$ and $a(n) = p_n$ for $n \ge 1$.

noncomputable def a (n : ℕ) : ℕ :=
  match n with
  | 0 => 0
  | n' + 1 => Nat.nth Nat.Prime n'

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 = 2 := by⊢ a 1 = 2
  change Nat.nth Nat.Prime 0 = 2⊢ Nat.nth Nat.Prime 0 = 2
  exact Nat.nth_prime_zero_eq_twoAll goals completed! 🐙

Value of the sequence a at 2.

@[category test, AMS 11]
theorem a_2 : a 2 = 3 := by⊢ a 2 = 3
  change Nat.nth Nat.Prime 1 = 3⊢ Nat.nth Nat.Prime 1 = 3
  exact Nat.nth_prime_one_eq_threeAll goals completed! 🐙

Value of the sequence a at 3.

@[category test, AMS 11]
theorem a_3 : a 3 = 5 := by⊢ a 3 = 5
  change Nat.nth Nat.Prime 2 = 5⊢ Nat.nth Nat.Prime 2 = 5
  exact Nat.nth_prime_two_eq_fiveAll goals completed! 🐙

Value of the sequence a at 4.

@[category test, AMS 11]
theorem a_4 : a 4 = 7 := by⊢ a 4 = 7
  change Nat.nth Nat.Prime 3 = 7⊢ Nat.nth Nat.Prime 3 = 7
  exact Nat.nth_prime_three_eq_sevenAll goals completed! 🐙

Conjecture from Thomas Ordowski (2023): $\log \log a(n+1) - \log \log a(n) < 1/n$ for $n > 0$.

@[category research open, AMS 11]
theorem conjecture (n : ℕ) (hn : 0 < n) :
    Real.log (Real.log (a (n + 1) : ℝ)) - Real.log (Real.log (a n : ℝ)) < 1 / (n : ℝ) := byn:ℕhn:0 < n⊢ Real.log (Real.log ↑(a (n + 1))) - Real.log (Real.log ↑(a n)) < 1 / ↑n
  sorryAll goals completed! 🐙

end OeisA40