Apples make potatoes sprout by their release of ethylene.
"Ethylene is a natural plant hormone released in the form of a gas. It triggers cells to degrade, fruit to turn softer and sweeter, leaves to droop, and seeds or buds to sprout."
Apples and pears are both high producers of it, potatoes and onions are both sensitive to it. There are more fruits and vegetables that either produce it in high amounts or react to it. This is why you don't want to store apples in your root cellar too close to your potatoes or onions, they will start sprouting and won't last as long.
Storing fruit that was picked too green by apples makes them ripen, that's how grocery stores ripen things like bananas that are shipped green, they gas them with ethylene.
List from Food Science: Ethylene
Ethylene producing foods:
apples, apricots, avocados, bananas (ripe), blueberries, cantaloupe, cherimoyas, cranberries, figs, green onions, guavas, grapes, honeydew, kiwifruit, mangoes, mangosteen, nectarines, papayas, passion fruit, peaches, pears, persimmons, plums, potatoes, prunes, quince, tomatoes
Ethylene sensitive foods:
asparagus, bananas (unripe), blackberries, broccoli, Brussels sprouts, cabbage, carrots, cauliflower, chard, cucumbers, eggplant, endive, garlic, green beans, kale, leafy greens, leeks, lettuce, okra, onions, parsley, peas, peppers, raspberries, spinach, squash, strawberries, sweet potatoes, watercress, watermelon
Useful trait to know about!