Sums of distinct divisors of factorials

For practical $N$ let $h(N)$ be the least $k$ such that every integer $1\le m\le N$ is a sum of at most $k$ distinct divisors of $N$. We prove $h(n!)\le (2\log 2+o(1))n\mathrm{/}\log n$. This improves the bounds of order $n\mathrm{/}(\log n{)}^{1\mathrm{/}2-\epsilon }$ established in Tenenbaum-Yokota’s Lemma 4 and Yokota’s 1995 knapsack note. We combine their decreasing greedy construction with the sharper factorial divisor-gap estimate of Berend-Harmse. Counting the steps separately below and above $\sqrt{n!}$, with the upper range handled through reciprocal divisors, retains the leading coefficient in the gap exponent and yields the explicit constant $2\log 2$.