On an integer’s infinitary divisors

Abstract

The notions of unitary divisor and biunitary divisor are extended in a natural fashion to give<italic>k-ary divisors</italic>, for any natural number<italic>k</italic>. We show that we may sensibly allow<italic>k</italic>to increase indefinitely, and this leads to<italic>infinitary divisors</italic>. The infinitary divisors of an integer are described in full, and applications to the obvious analogues of the classical perfect and amicable numbers and aliquot sequences are given.