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balanced_primes_order

Specification

Let $p_k$ be the $k$-th prime number. Are there infinitely many $n$ such that $p_n = \dfrac{\sum_{i = 1} ^ k p_{n - i} + p_{n + i}}{2*k}$?

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Lean 4 Statement 
theorem balanced_primes_order :
    answer(sorry) ↔ ∀ k > 0, {n : ℕ | k ≤ n ∧ 2 * k * Nat.nth Prime n = ∑ i ∈ .Ioc 0 k,
      ((n - i).nth Prime + (n + i).nth Prime)}.Infinite
ID: Wikipedia__BalancedPrimes__balanced_primes_order
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