Are there odd amicable numbers not divisible by three?

Quasi-Amicable Numbers are Rare

Define a quasi-amicable pair as a pair of distinct natural numbers each of which is the sum of the nontrivial divisors of the other, e.g., {48, 75}. Here nontrivial excludes both 1 and the number itself. Quasi-amicable pairs have been studied (primarily empirically) by Garcia, Beck and Najar, Lal and Forbes, and Hagis and Lord. We prove that the set of n belonging to a quasi-amicable pair has a...

Construction of Number Fields of Odd Degree with Class Numbers Divisible by Three, Five or by Seven

On amicable numbers

Let A (x) denote the set of integers n≤ x that belong to an amicable pair. We show that #A (x)≤ x/e √ logx for all sufficiently large x.

Breeding amicable numbers in abundance. II

In a first article of this title, new procedures were described to compute many amicable numbers by “breeding” them in several generations. An extensive computer search was later performed (in 1988), and demonstrated the remarkable effectiveness of this breeding method: the number of known amicable pairs was easily quadrupled by this search. As we learnt recently (1999) from the internet, Peder...

On Amicable Numbers With Different Parity

In this paper we provide a straightforward proof that if a pair of amicable numbers with different parity exists (one number odd and the other one even), then the odd amicable number must be a perfect square, while the even amicable number has to be equal to the product of a power of 2 and an odd perfect square. 1 Introductory remarks An amicable pair (M,N) consists of two integers M , N for wh...