Problem 254

Sums of Digit Factorials

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?

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