A remark on sociable numbers of odd order

Write s(n) for the sum of the proper divisors of the natural number n. We call n sociable if the sequence n, s(n), s(s(n)), … is purely periodic; the period is then called the order of sociability of n. The ancients initiated the study of order 1 sociables (perfect numbers) and order 2 sociables (amicable numbers), and investigations into higher-order sociable numbers began at the end of the 19th century. We show that if k is odd and fixed, then the number of sociable n⩽x of order k is bounded by as x→∞. This improves on the previously best-known bound of , due to Kobayashi, Pollack, and Pomerance.