map 16-bit integers to RGB colors with monotonic relative luminosity
Since you want monotonically increasing luminance, it's probably best to work in a color space that separates luma from chroma, such as a Y'UV space. You can assign values in Y'UV and then convert it to RGB using the matrix at that link.

Start by setting $Y' = N / (2^{16} - 1)$, where N is your input 16-bit integer. Now you just need to make up some function to use for the U and V components.

It wasn't clear to me if you want the chroma part of the color to vary continuously with the input N, or be pseudo-random. For a continuous curve, you could just have it be a circle in UV space:
$$
U = 0.5 \cos(2\pi N/2^{16}) \\
V = 0.5 \sin(2\pi N/2^{16})
$$
or whatever other curve you feel like coming up with. For a pseudo-random chroma, you could use a hash function, or a Halton sequence or another low-discrepancy sequence.