1995
A new method for finding amicable pairs
Publication
Publication
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 19431993, 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.
Additional Metadata  

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

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