How many hours a week does a 20 year old have to spend in the sun, without a shirt or sunscreen, to get enough vitamin d?

At most 1/2 hour. The skin rapidly produces sufficient vitamin d on sun exposure. In 1/2 hour you have made 10, 000 units of vitamin d3, quite adequate for any known needs. After than you run the risk of sunburn, skin changes and possibly cancer later in light (if done often) although you might tan with repeated exposure (if that is one of your goals).
~10 minutes a day. The amount of time needed to synthesize an adequate amount of vit d varies by skin type and latitude. 20-30 minutes of exposure to the arms and legs 3 times a week is usually sufficient. With the shirt off, more skin is exposed. This needs to be coupled with adequate dietary intake. If there is a concern of vit d deficiency, see your doctor.