I want roots (tubers), not foliage!
I keep hearing this old wives tale, but one must remember that it is the leaves (foliage) that manufacture the food that is stored in the tuber. Those leaves need nitrogen to make that food too. In my experience, with any plant, you will get larger and more fruit if you first get larger and more leaves. I will extend this to the root crops too.
Its pretty easy to make the test in your own garden. Make two plots for any variety you want to test this on, and fertilize one and not the other.
I don't make normally make recommendations on types of fertilizer, because I know many people are pretty picky about what they will allow in their gardens. So, I am sticking to what I said earlier; "One more thing I will say is that potatoes respond well to nitrogen. Good fertile soil is a must to get the large tubers."