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

Question

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

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