Let $sigma(x)$ denote the sum of all divisors of the (positive) integer x. An amicable pair is a pair of integers $(m,n)$ with $m<n$ such that $sigma(m)=sigma(n)=m+n$. The smallest amicable pair is $(220,284)$. A new method for finding amicable pairs is presented, based on the following observation of ErdH{os: For given s, let $x_1, x_2, dots$ be solutions of the equation $sigma(x)=s$, then any pair $(x_i,x_j)$ for which $x_i+x_j=s$ is amicable. The problem here is to find numbers s for which the equation $sigma(x)=s$ has many solutions. From inspection of tables of known amicable pairs and their pair sums one learns that certain smooth numbers s (i.e., numbers with only small prime divisors) are good candidates. With the help of a precomputed table of $sigma(p^e)$-values, many solutions of the equation $sigma(x)=s$ were found by checking divisibility of s by the tabled $sigma$-values in a recursive way. In the set of solutions found, pairs were traced which sum up to s. From 1850 smooth numbers s satisfying $4times10^{11<s<10^{12$ we found 116 new amicable pairs with this algorithm. After the submission of this paper to the Vancouver Conference Mathematics of Computation 1943--1993, the computations have been extended and yielded many more new amicable pairs. In particular, the first quadruple of amicable pairs with the same pair sum (namely $16!$) was found. A list is given of 587 amicable pairs with smaller member between $2.01times10^{11$ and $10^{12$, of which 565 pairs seem to be new.

Department of Numerical Mathematics [NM]
Large-scale computing

te Riele, H. (1995). A new method for finding amicable pairs. Department of Numerical Mathematics [NM]. CWI.