erdos_394_factorial_gap_conjecture

They ask about the behaviour of $t_{n-3}(n!)$ and also ask whether, for infinitely many $n$, $t_k(n!)< t_{k-1}(n!)-1$ for all $1\leq k < n$.

theorem erdos_394_factorial_gap_conjecture :
    answer(sorry) ↔
      Set.Infinite { n : ℕ | ∀ k, 2 ≤ k → k < n →
      t k (n !) < t (k - 1) (n !) - 1 }

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