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import FormalConjecturesUtil$a(n)$ = the number of values of $k <= 10^n$ such that $\sqrt{k(k+1)(k+2)(k+3)+1}$ is prime
Since $\sqrt{k(k+1)(k+2)(k+3)+1} = k^2 + 3k + 1$, $a(n) = #{k \in \mathbb{N} \mid 1 \le k \le 10^n \land (k^2 + 3k + 1) \text{ is prime} }.$
References:
namespace OeisA115366open Filter Real Topology
The primary defining sequence a.
$a(n) = #{k \in \mathbb{N} \mid 1 \le k \le 10^n \land (k^2 + 3k + 1) \text{ is prime} }.$
def a (n : ℕ) : ℕ :=
Finset.card <|
Finset.filter
(fun k : ℕ => Nat.Prime (k ^ 2 + 3 * k + 1))
(Finset.Icc 1 (10 ^ 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 = 9 := ⊢ a 1 = 9 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 50 := ⊢ a 2 = 50 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 313 := ⊢ a 3 = 313 All goals completed! 🐙Conjecture: $a(n)/A006880(n) \rightarrow 1.77...$ where A006880(n) is the number of primes $\le 10^n$.
@[category research open, AMS 11]
theorem conjecture :
∃ L : ℝ,
Tendsto (fun n : ℕ => (a n : ℝ) / (Nat.primeCounting' (10 ^ n) : ℝ)) atTop (nhds L) ∧
1.77 ≤ L ∧ L ≤ 1.78 := ⊢ ∃ L, Tendsto (fun n ↦ ↑(a n) / ↑(10 ^ n).primeCounting') atTop (𝓝 L) ∧ 1.77 ≤ L ∧ L ≤ 1.78
All goals completed! 🐙end OeisA115366