A list initially contains the numbers $2,3,\dots ,n$.
At each round, every number in the list is divided by its smallest prime factor. Then the product of these smallest prime factors is added to the list as a new number. Finally, all numbers that become $1$ are removed from the list.

For example, below are the first three rounds for $n=5$: $$[2,3,4,5]\stackrel{(1)}{\to }[2,60]\stackrel{(2)}{\to }[30,4]\stackrel{(3)}{\to }[15,2,4].$$ Let $S(n,m)$ be the sum of all numbers in the list after $m$ rounds.
For example, $S(5,3)=15+2+4=21$. Also $S(10,100)=257$.

Find $S({10}^4,{10}^{16})$. Give your answer modulo $1234567891$.