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import FormalConjecturesUtilConjectures associated with A109909
$a(n)$ = number of primes of the form $k(n-k)-1$.
References:
namespace OeisA109909$a(n)$ is the number of primes of the form $k(n-k)-1$. This counts the number of distinct prime values in the set ${k(n-k)-1 | 1 \le k < n}$.
def a (n : ℕ) : ℕ :=
(Finset.Ico 1 n).image (fun k => k * (n - k) - 1)
|>.filter Nat.Prime
|>.card@[category test, AMS 11]
theorem a_1 : a 1 = 0 := ⊢ a 1 = 0 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 0 := ⊢ a 2 = 0 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 0 := ⊢ a 3 = 0 All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 2 := ⊢ a 4 = 2 All goals completed! 🐙@[category test, AMS 11]
theorem a_5 : a 5 = 2 := ⊢ a 5 = 2 All goals completed! 🐙Conjecture: $a(n) > 0$ for $n > 3$.
@[category research open, AMS 11]
theorem conjecture : ∀ n > 3, a n > 0 := ⊢ ∀ n > 3, OeisA109909.a n > 0
All goals completed! 🐙end OeisA109909