jmc

We are given that $x(x+1)=506$, so $x^2+x=506$, which means $x^2+x-506=0$. The prime factorization of $506$ is $2\cdot 11\cdot 23$, so we see that the quadratic factors as $(x+23)(x-22)=0$. The positive solution is $x=22$, so the two numbers are 22 and 23. Their sum is $22+23=\boxed{45}$.