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import FormalConjecturesUtilA GCD-Driven Sequence with Universal Jumps
OEIS A260194 begins with three ones and then adds the greatest common divisor of the current term and the term two places earlier. The open question asks whether every positive integer occurs as an adjacent difference.
References:
namespace OeisA260194OEIS A260194, shifted so that Lean index zero is the first OEIS term, with recurrence $a(n+3) = a(n+2) + \gcd(a(n+2),a(n))$ and initial values $a(0)=a(1)=a(2)=1$.
def a : ℕ → ℕ
| 0 => 1
| 1 => 1
| 2 => 1
| n + 3 => a (n + 2) + Nat.gcd (a (n + 2)) (a n)@[category test, AMS 11]
theorem a_0 : a 0 = 1 := ⊢ a 0 = 1 All goals completed! 🐙@[category test, AMS 11]
theorem a_1 : a 1 = 1 := ⊢ a 1 = 1 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 1 := ⊢ a 2 = 1 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 2 := ⊢ a 3 = 2 All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 3 := ⊢ a 4 = 3 All goals completed! 🐙@[category test, AMS 11]
theorem a_5 : a 5 = 4 := ⊢ a 5 = 4 All goals completed! 🐙@[category test, AMS 11]
theorem a_6 : a 6 = 6 := ⊢ a 6 = 6 All goals completed! 🐙@[category test, AMS 11]
theorem a_7 : a 7 = 9 := ⊢ a 7 = 9 All goals completed! 🐙@[category test, AMS 11]
theorem a_8 : a 8 = 10 := ⊢ a 8 = 10 All goals completed! 🐙@[category test, AMS 11]
theorem a_9 : a 9 = 12 := ⊢ a 9 = 12 All goals completed! 🐙Does every positive integer occur as a difference in this sequence?
@[category research open, AMS 11]
theorem conjecture :
answer(sorry) ↔ ∀ d > 0, ∃ n, a (n + 1) = a n + d := ⊢ True ↔ ∀ d > 0, ∃ n, OeisA260194.a (n + 1) = OeisA260194.a n + d
All goals completed! 🐙end OeisA260194