The difference between the squares of two consecutive integers is 408235. The sum of the numbers is

Let's denote the two consecutive integers as n and n + 1. 

The difference between their squares is given by:

(n + 1)^2 - n^2 = 408235

Simplifying this equation, we get:

n^2 + 2n + 1 - n^2 = 408235
2n + 1 = 408235

Solving for n, we get:

n = (408235 - 1) / 2 = 204117

The next consecutive integer is n + 1 = 204118

So, the sum of these two consecutive numbers is:

204117 + 204118 = 408235

Therefore, the correct answer is option 4: 408235.