Eccoci.

Define $f(n)$ as the sum of the factorials of the digits of $n$. For example, $f(342) = 3! + 4! + 2! = 32$.

Define $sf(n)$ as the sum of the digits of $f(n)$. So $sf(342) = 3 + 2 = 5$.

Define $g(i)$ to be the smallest positive integer $n$ such that $sf(n) = i$. Though $sf(342)$ is $5$, $sf(25)$ is also $5$, and it can be verified that $g(5)$ is $25$.

Define $sg(i)$ as the sum of the digits of $g(i)$. So $sg(5) = 2 + 5 = 7$.

Further, it can be verified that $g(20)$ is $267$ and $\sum sg(i)$ for $1 \le i \le 20$ is $156$.

What is $\sum sg(i)$ for $1 \le i \le 150$?