Amicable numbers

A pair of natural numbers each one of which is equal to the sum of the proper divisors of the other, i.e. of the divisors other than the number itself. This definition is found already in Euclid's Elements and in the works of Plato. One pair of such numbers only — 220 and 284 — was known to the ancient Greeks; the sums of their divisors are equal, respectively, to

$$1+2+4+5+10+11+20+22+44+55+110=284,$$

$$1+2+4+71+142=220.$$

L. Euler discovered about 60 pairs of amicable numbers, while the use of electronic computers yielded a few hundreds of such numbers. It is not known, however, if there exists a pair of amicable numbers one of which is even while the other is odd.

For some very large pairs see [Ri].

References

[Ri]	H.J.J. te Riele, "New very large amicable pairs" , Proc. Number Theory Noordwijkerhout, 1983 , Lect. notes in math. , 1068 , Springer (1983) pp. 210–215