Amicable numbers

2020 Mathematics Subject Classification: Primary: 11A25 [MSN][ZBL]

A pair of natural numbers each one of which is equal to the sum of the aliquot 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 59 pairs of amicable numbers, while the use of electronic computers yielded a few hundreds of such numbers: for some very large examples see [Ri]. It is not known whether the number of amicable pairs is finite or infinite, nor whether there exists a pair of amicable numbers one of which is even while the other is odd: although it is known that the sum of the reciprocals of the amicable numbers converges [Po1]. If $A(x)$ denotes the number of integers $\le x$ that belong to an amicable pair, then it is known that $$ A(x) \le x \exp(-\sqrt{\log x}) $$ for all sufficiently large $x$, see [Po2].

See also Aliquot sequences.

References

[Po1]	Carl Pomerance, On the distribution of amicable numbers. II J. Reine Angew. Math. 325 (1981) 183-188 Zbl 0448.10007
[Po2]	Carl Pomerance, On amicable numbers in "Analytic number theory. In honor of Helmut Maier’s 60th birthday": Springer (2015) ISBN 978-3-319-22239-4 Zbl 06569787
[Ri]	H.J.J. te Riele, New very large amicable pairs in "Proc. Number Theory Noordwijkerhout, 1983" , Lecture notes in mathematics 1068 Springer (1983) pp. 210–215 Zbl 0539.10008