View source for Amicable numbers

{{TEX|done}}{{MSC|11A25}}

A pair of natural numbers each one of which is equal to the sum of the [[aliquot divisor]]s 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 {{Cite|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.  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 {{Cite|Po}}.

See also [[Aliquot sequence]]s.

====References====

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|valign="top"|{{Ref|Po}}||valign="top"| 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}}
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|valign="top"|{{Ref|Ri}}||valign="top"| 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 {{ZBL|0539.10008}}
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