1. FormalConjectures
  2. OEIS
  3. «110854»
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Conjectures associated with A110854

$a(n) = \mathrm{prime}(2n+2) - \mathrm{prime}(2n+1) - \mathrm{prime}(2n) + \mathrm{prime}(2n-1)$, where $\mathrm{prime}(k)$ is the $k$-th prime number.

References:

namespace OeisA110854open Nat

The primary defining sequence a. $a(n)$ is $\mathrm{prime}(2n+2) - \mathrm{prime}(2n+1) - \mathrm{prime}(2n) + \mathrm{prime}(2n-1)$.

noncomputable def a (n : ℕ) : ℤ := let p (k : ℕ) : ℤ := (Nat.nth Nat.Prime (k - 1)).cast if n = 0 then 0 else p (2 * n + 2) - p (2 * n + 1) - p (2 * n) + p (2 * n - 1)h1:Nat.Prime 13⊢ nth Nat.Prime 5 = 13 All goals completed! 🐙h1:Nat.Prime 17⊢ nth Nat.Prime 6 = 17 All goals completed! 🐙h1:Nat.Prime 19⊢ nth Nat.Prime 7 = 19 All goals completed! 🐙h1:Nat.Prime 23⊢ nth Nat.Prime 8 = 23 All goals completed! 🐙h1:Nat.Prime 29⊢ nth Nat.Prime 9 = 29 All goals completed! 🐙

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 := ⊢ a 0 = 0 All goals completed! 🐙@[category test, AMS 11] theorem a_1 : a 1 = 1 := ⊢ a 1 = 1 ⊢ ↑(nth Nat.Prime 3) - ↑(nth Nat.Prime 2) - ↑(nth Nat.Prime 1) + ↑(nth Nat.Prime 0) = 1 All goals completed! 🐙⊢ ↑13 - ↑11 - ↑7 + ↑5 = 0 All goals completed! 🐙⊢ ↑19 - ↑17 - ↑13 + ↑11 = 0 All goals completed! 🐙⊢ ↑29 - ↑23 - ↑19 + ↑17 = 4 All goals completed! 🐙

Do the absolute values cover A004275? A004275 is $1$ together with the nonnegative even numbers. The conjecture asks whether every member of A004275 occurs as $|a(n)|$ for some term of the sequence.

@[category research open, AMS 11] theorem conjecture : ∀ d : ℕ, (d = 1 ∨ Even d) → ∃ n > 0, d = (a n).natAbs := ⊢ ∀ (d : ℕ), d = 1 ∨ Even d → ∃ n > 0, d = (OeisA110854.a n).natAbs All goals completed! 🐙end OeisA110854