A remark on sociable numbers of odd order

作者：Pollack Paul^*

来源：Journal of Number Theory, 2010, 130(8): 1732-1736.

DOI：10.1016/j.jnt.2010.03.011

• Sociable number
• Sociable chain
• Aliquot cycle
• Aliquot part

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 x/(logx)(1+o(1)) as x -> infinity. This improves on the previously best-known bound of x/(log logx)(1/2+o(1)), due to Kobayashi. Pollack, and Pomerance.

• 出版日期2010-8