Factorial Sums

Find the sum of all positive integers mm such that 2m2^m can be expressed as sums of four factorials (of positive integers).

Details and assumptions

The number n! n!, read as n factorial, is equal to the product of all positive integers less than or equal to nn. For example, 7!=7×6×5×4×3×2×1 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1.

The factorials do not have to be distinct. For example, 24=162^4=16 counts, because it equals 3!+3!+2!+2!3!+3!+2!+2!.

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