Let $f(n)$ be the largest prime factor of $n$.

Let $g(n)=f(n)+f(n+1)+f(n+2)+f(n+3)+f(n+4)+f(n+5)+f(n+6)+f(n+7)+f(n+8)$, the sum of the largest prime factor of each of nine consecutive numbers starting with $n$.

Let $h(n)$ be the maximum value of $g(k)$ for $2\le k\le n$.

You are given:

• $f(100)=5$
• $f(101)=101$
• $g(100)=409$
• $h(100)=417$
• $h({10}^9)=4896292593$

Find $h({10}^{16})$.