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import FormalConjecturesUtil
namespace OeisA102847
The primary defining sequence a.
$a(0)=1$, $a(n) = a(n-1)^2 + 2$.
def a : ℕ → ℕ
| 0 => 1
| n + 1 => (a n) ^ 2 + 2Term theorems verifying the first few values of the sequence against the official OEIS b-file
@[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 = 3 := ⊢ a 1 = 3 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 11 := ⊢ a 2 = 11 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 123 := ⊢ a 3 = 123 All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 15131 := ⊢ a 4 = 15131 All goals completed! 🐙Prime for $a(1) = 3$, $a(2) = 11$, $a(4) = 15131$; semiprime for $a(3) = 123 = 3 * 41$, $a(5) = 228947163 = 3 * 76315721$. $a(6)$, added by Jonathan Vos Post, has 4 prime factors. $a(7) = 41 * 811^2 * 106693969 * 317171188688357726699 * 8272236925540996054440172449761$. When is the next prime in the sequence?
@[category research open, AMS 11]
theorem conjecture : answer(sorry) = sInf {n : ℕ | 4 < n ∧ (a n).Prime} := ⊢ sorry = sInf {n | 4 < n ∧ Nat.Prime (a n)}
All goals completed! 🐙end OeisA102847