Finding Closed-form expression for the biggest partition

Question

Let us define the following series: $A_n$ is the greatest number, $k$, that satisfies:

For every set of natural numbers $B$ that satisfies:

• $\max (B)\leq (n-1)! $
• $\text{sum}(B)\geq n!$

there exists a partition of B into disjoint subsets $B_1, B_2, \dots B_k$ such that $\forall i\leq k. \text{sum}(B_i) \geq (n-1)!$

1. Is there some simple closed-form expression for $A_n$?
2. Is it true that $\forall n\geq 1. A_n\geq \lfloor n/2\rfloor$?
3. Is there any better lower bound?

Edit: Clearly, the first values are: $A_1=1, A_2=2, A_3 =3$, but I couldn't compute greater values.