Corn is a heavy nitrogen feeder and beans are a good succession crop. I plant too closely to plant anything else with the corn so I can maximize the number of corn plants in the space. I plant 8-12 inches all around. I can plant up to three successive corn crops but I follow the corn with a nutrient scavenger like Asian greens in the fall. If I do plant legumes for nitrogen fixing, I have to inoculate the beans. I have grown cowpeas as a cover crop and nitrogen fixer and that could be grown with the corn if the corn is spaced at least 12 inches apart. Cowpeas don't do a lot of climbing so they should not pull down the stalks. If they are grown for the pods they don't fix as much nitrogen.
If you are trying to start an organic garden, it takes a while to prep it. It does take about 3 years for it to get up to speed. You may have a good soil to start with and can get away with a decent crop the first time out, but it is better to put the time into really preparing the beds well for the long run. Add the compost and manure and build the soil. Organic prep is usually better started 2-6 months before. Cover crops like buckwheat and cowpeas add green manure to help the sucessive crop. It is better to start with low nitrogen feeders. You will have to supplement an organic garden weekly with organic fertilizer, AACT, or fish emulsion to get enough nitrogen to support the growth of young plants.
Arugula, beans, beets, carrots, chicory, collard greens, endive, escarole, fava beans, herbs (most kinds), kale, parsnip, peas, Swiss chard
Artichoke, basil, cilantro, sweet corn, cucumbers, eggplant, garlic, lettuce, okra, peppers (small-fruited), potatoes, pumpkin, radish, rutabaga, scallions, squash, tomatoes, watermelon, zucchini
Asparagus, broccoli, Brussels sprouts, cabbage, cantaloupe/honeydew, cauliflower, celery/celeriac, kohlrabi, leeks, onions, peppers (large-fruited), spinach, turnips