Shadows of uniform set families

For a finite family $\mathcal F$ of distinct $k$-element sets, let $d(\mathcal F)$ be the collection of all $(k-1)$-element sets contained in at least one member of $\mathcal F$, and write $D(\mathcal F)=|d(\mathcal F)|$. Let $M(k)$ be the largest positive integer that is not equal to $D(\mathcal F)$ for any finite family $\mathcal F$ of distinct $k$-element sets.

For fixed $k$ and $t$, let $L_k(t)$ be the smallest integer $L$ such that every integer from $L$ through $tk$ is equal to $D(\mathcal F)$ for some $t$-member family $\mathcal F$ of distinct $k$-element sets. Let $\tau(k)$ be the largest $t$ in $\{1,\ldots,k+1\}$ with $L_k(t)\ge tk-k+2$, and let $h(k)$ be the least integer $j\ge0$ such that $\binom{j+3}{2}\ge \tau(k)$.

Among the integers $k$ with $374\le k\le25000$ for which $\gcd(M(k),\tau(k)(h(k)+1))=1$, what is the sum of those $k$?