Generate limited size colour palette for image data [textures/images]

How can I generate a palette of 16 colours and a palette of 256 colours from an image that will be different sizes.

Is there any function or formula out there that can do this?

I've checked this post, but I don't want to use a massive library, which I'm guessing Qt is. What would be a good algorithm to convert an image to a limited size color palette

And this post doesn't take into account a maximum count of colours in the palette. Generate colour palette from an image