## Sums of Digit Factorials

### Problem 254

Published on Friday, 4th September 2009, 05:00 pm; Solved by 634; Difficulty rating: 75%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 ∑ sg(`i`) for 1 ≤ `i` ≤ 20 is 156.

What is ∑ sg(`i`) for 1 ≤ `i` ≤ 150?