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import FormalConjecturesUtilNumber of decompositions of $2n+1$ into $2p+q$, where $p$ and $q$ are both odd primes
References:
namespace OeisA103151
The primary defining sequence a.
a n is the number of decompositions of 2n+1 into 2p+q, where p and q are both odd primes.
def a (n : ℕ) : ℕ :=
-- We count the number of odd primes $p$ that satisfy the constraints.
-- The range $p \le n$ is sufficient, as larger $p$ would make $q \le 1$, which is not prime.
Finset.card (Finset.filter (fun p : ℕ =>
-- p must be an odd prime.
p.Prime ∧
p ≠ 2 ∧
-- Ensure the expression for $q$ is positive, so Nat subtraction is well-defined for prime $q$.
2 * p < 2 * n + 1 ∧
-- q = 2n+1 - 2p must be prime (and is automatically odd).
Nat.Prime (2 * n + 1 - 2 * p)
) (Finset.range (n + 1)))Term theorems verifying the first few values of the sequence against the official OEIS b-file
@[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 = 1 := ⊢ a 4 = 1 All goals completed! 🐙@[category test, AMS 11]
theorem a_5 : a 5 = 1 := ⊢ a 5 = 1 All goals completed! 🐙Conjecture: all items for $n \ge 4$ are greater than or equal to $1$. This is a stronger conjecture than the Goldbach conjecture.
@[category research open, AMS 11]
theorem conjecture (n : ℕ) (hn : n ≥ 4) : a n ≥ 1 := n:ℕhn:n ≥ 4⊢ a n ≥ 1
All goals completed! 🐙end OeisA103151