KöMaL 2018 B4932

The Great Bestiary of Wonderland features a dragon for every week of the year. All dragons have different ages. The youngest dragon, named Aloysius has 13 heads. The second youngest one, Bartholomeus has 14 heads, ... (and so on, each dragon in the order of their ages has one more head than the previous one). The oldest dragon, Zebulon has 64 heads. Wonderland monks are writing the Giant Codex of Dragon Tales. A tale may only be included in the Codex if the total number of heads of all the dragons in the tale is exactly 1001. For every pair of tales, the sets of dragons mentioned in the tales are different. Which of the 13-headed Aloysius and the 14-headed Bartholomeus will appear in more tales when the monks are finished with writing down all possible tales?