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.

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CWI
Department of Numerical Mathematics [NM]
Large-scale computing

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