Multiamicable Numbers

Abstract

Abstract.Multiamicable numbers are a natural generalization of amicable numbers: two numbers form a multiamicable pair if the sum of the proper divisors of each is a multiple of the other. Many other generalizations have been considered in the past. This paper reviews those earlier generalizations and gives examples and properties of multiamicable pairs. It includes a proof that the set of all multiamicable numbers has density 0. Generalizations of amicable numbersTwo natural numbers are said to be amicable if the sum of the proper divisors of each of them equals the other. Thus, where o denotes the positive divisor sum function, m and n are amicable if (1) o(m) -m = n and o(n)-n -m. This condition can be abbreviated to a(m) = a(n) = m + n . We assume that m ^ n . (If m = n , then m is perfect.) It is usual to order pairs of amicable numbers according to the size of the smaller member. The smallest pair of amicable numbers, known to the Pythagoreans, is 220 and 284. Many of the classical mathematicians, such as Fermât, Mersenne, Descartes, Legendre, and particularly Euler, studied amicable numbers. Over fifty thousand pairs are now known, and many techniques are known for generating new pairs from old ones, although these will not always be successful. It has not been determined, however, whether there are infinitely many pairs of amicable numbers.