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import FormalConjecturesUtil
namespace OeisA113609A number $n$ is an "OEIS prime power" (for the context of A113609's definition) if $n=1$ or $n$ is a standard prime power.
def IsOeisPrimePower (n : ℕ) : Prop := n = 1 ∨ IsPrimePow ninstance DecidableIsOeisPrimePower (n : ℕ) : Decidable (IsOeisPrimePower n) := n:ℕ⊢ Decidable (IsOeisPrimePower n)
n:ℕ⊢ Decidable (n = 1 ∨ IsPrimePow n)
All goals completed! 🐙
The primary defining sequence a.
$a(n)$ is the number of prime powers $q<=n$ such that also $q+2$ is a prime power.
$$a(n) = \operatorname{card} {q \in \mathbb{N} \mid 1 \le q \le n \land P(q) \land P(q+2) }$$
def a (n : ℕ) : ℕ :=
Finset.card $ (Finset.range (n + 1)).filter fun q =>
IsOeisPrimePower q ∧ IsOeisPrimePower (q + 2) ∧ q ≥ 1@[category test, AMS 11]
theorem a_1 : a 1 = 1 := ⊢ a 1 = 1 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 2 := ⊢ a 2 = 2 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 3 := ⊢ a 3 = 3 All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 3 := ⊢ a 4 = 3 All goals completed! 🐙@[category test, AMS 11]
theorem a_5 : a 5 = 4 := ⊢ a 5 = 4 All goals completed! 🐙(25,27) is the smallest pair of prime powers (q,q+2) such that both q and q+2 are not primes, conjecture: there are more (but not < 10^6).
@[category research open, AMS 11]
theorem conjecture :
answer(sorry) ↔ ∃ q ≥ 1000000,
IsOeisPrimePower q ∧ IsOeisPrimePower (q + 2) ∧
¬ q.Prime ∧ ¬ (q + 2).Prime := ⊢ True ↔ ∃ q ≥ 1000000, IsOeisPrimePower q ∧ IsOeisPrimePower (q + 2) ∧ ¬Nat.Prime q ∧ ¬Nat.Prime (q + 2)
All goals completed! 🐙end OeisA113609