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import FormalConjecturesUtilPrimality and primitive root property from divisibility $n \mid (a(n-1) + 2^{n-2})$
$a(n) = \frac{2^{n-1}}{n!} \prod_{k=1}^{n-1} (2^k-1)$. The sequence $a(n)$ is composed of natural numbers, thus we define it as a function $\mathbb{N} \to \mathbb{N}$.
References:
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
namespace OeisA91669open Nat BigOperators$a(n) = \frac{2^{n-1}}{n!} \prod_{k=1}^{n-1} (2^k-1)$. The sequence $a(n)$ is composed of natural numbers, thus we define it as a function $\mathbb{N} \to \mathbb{N}$.
def a (n : ℕ) : ℕ :=
if n = 0 then 0 -- Sequence is defined for n >= 1.
else
let n_pred : ℕ := n.pred
-- The numerator of the expression. Both factors are in ℕ.
let numerator : ℕ := (2 ^ n_pred) * (Finset.Ico 1 n).prod (fun k => 2 ^ k - 1)
-- The denominator is $n!$.
let denominator : ℕ := n.factorial
-- The division is exact, since the result is an integer sequence.
numerator / denominator@[category test, AMS 11]
lemma a_1 : a 1 = 1 := ⊢ a 1 = 1 All goals completed! 🐙@[category test, AMS 11]
lemma a_2 : a 2 = 1 := ⊢ a 2 = 1 All goals completed! 🐙@[category test, AMS 11]
lemma a_3 : a 3 = 2 := ⊢ a 3 = 2 All goals completed! 🐙@[category test, AMS 11]
lemma a_4 : a 4 = 7 := ⊢ a 4 = 7 All goals completed! 🐙@[category test, AMS 11]
lemma a_5 : a 5 = 42 := ⊢ a 5 = 42 All goals completed! 🐙Conjecture (for $n > 2$): if $n \mid a(n-1) + 2^{n-2}$, then $n$ is a prime with primitive root 2 (A001122). - Amiram Eldar and Thomas Ordowski, Jan 19 2020
A formal proof has been found with the methods described in arxiv/2605.22763.
@[category research solved, AMS 11, formal_proof using formal_conjectures at
"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/91669.wip.lean#L250"]
theorem prime_and_primitive_root_of_dvd (n : ℕ) (hn : n > 2) : n ∣ (a (n - 1) + 2 ^ (n - 2)) →
Nat.Prime n ∧ IsPrimitiveRoot (2 : ZMod n) (Nat.totient n) := n:ℕhn:n > 2⊢ n ∣ a (n - 1) + 2 ^ (n - 2) → Nat.Prime n ∧ IsPrimitiveRoot 2 (φ n)
All goals completed! 🐙end OeisA91669