If x is the smallest of the three, then the next two integers are x + 2 and x + 4.
"Twice the largest is 20 less than the sum of all three" translates to
2 (x + 4) = (x + (x + 4) + (x + 6)) - 20
Solve for x :
2x + 8 = 3x - 10
x = 18
Then the three numbers are {18, 20, 22}.