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import FormalConjecturesUtilNumerator of $\sum_{k=1}^n 2^k/k$.
Conjecture: for $n > 3$, $\textrm{numerator}(-2/n + \sum_{k=1}^{n} \frac{2^k}{k}) == 0 (\textrm{mod} n^2)$ if and only if n is prime.
References:
namespace OeisA108866
The primary defining sequence a.
a n is the numerator of $\sum_{k=1}^n \frac{2^k}{k}$.
noncomputable def a (n : ℕ) : ℕ :=
(∑ i ∈ Finset.range n, (2 : ℚ) ^ (i + 1) / ((i + 1) : ℚ)).num.natAbslocal macro "eval_a" : tactic => `(tactic| (delta a; norm_num))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 = 2 := ⊢ a 1 = 2 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 4 := ⊢ a 2 = 4 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 20 := ⊢ a 3 = 20 All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 32 := ⊢ a 4 = 32 All goals completed! 🐙The rational number inside the numerator function in the conjecture. $$ -\frac{2}{n} + \sum_{k=1}^n \frac{2^k}{k} $$
noncomputable def ratExpression (n : ℕ) : ℚ :=
if n > 0 then
(-2 : ℚ) / (n : ℚ) + ∑ i ∈ (Finset.range n), (2 : ℚ) ^ (i + 1) / ((i + 1) : ℚ)
else
0Conjecture: for $n > 3$, $\textrm{numerator}(-2/n + \sum_{k=1}^{n} \frac{2^k}{k}) == 0 (\textrm{mod} n^2)$ if and only if n is prime.
@[category research open, AMS 11]
theorem conjecture {n : ℕ} (hn : n > 3) :
(ratExpression n).num ≡ 0 [ZMOD (n^2 : ℤ)] ↔ n.Prime := n:ℕhn:n > 3⊢ (ratExpression n).num ≡ 0 [ZMOD ↑n ^ 2] ↔ Nat.Prime n
All goals completed! 🐙end OeisA108866