Cuboid layers

The minimum number of cubes to cover every visible face on a cuboid measuring $3$ x $2$ x $1$ is twenty-two.

If we then add a second layer to this solid it would require forty-six cubes to cover every visible face, the third layer would require seventy-eight cubes, and the fourth layer would require one-hundred and eighteen cubes to cover every visible face.

However, the first layer on a cuboid measuring $5$ x $1$ x $1$ also requires twenty-two cubes; similarly the first layer on cuboids measuring $5$ x $3$ x $1$, $7$ x $2$ x $1$, and $11$ x $1$ x $1$ all contain forty-six cubes.

We shall define C(n) to represent the number of cuboids that contain n cubes in one of its layers. So C($22$) = $2$, C($46$) = $4$, C($78$) = $5$, and C($118$) = $8$.

It turns out that $154$ is the least value of n for which C(n) = $10$.

Find the least value of n for which C(n) = $1000$.