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%I #65 Jul 15 2015 18:02:24
%S 220,284,1184,1210,2620,2924,5020,5564,6232,6368,10744,10856,12285,
%T 14595,17296,18416,66928,66992,67095,71145,63020,76084,69615,87633,
%U 79750,88730,100485,124155,122368,123152,122265,139815,141664,153176,142310,168730,171856,176336,176272,180848,185368,203432,196724,202444,280540,365084,308620,389924
%N Amicable pairs (x < y) ordered by nondecreasing sum (x + y) and then by increasing x.
%C A pair of numbers x and y is called amicable if the sum of the proper divisors (or aliquot parts) of either one is equal to the other.
%C By definition a property of the amicable pair (x, y) is that x + y = sigma(x) = sigma(y).
%C The amicable pairs (x < y) are adjacent to each other in the list.
%C Also A260086 and A260087 interleaved.
%C Another version (A259180) lists the amicable pairs (x < y) ordered by increasing x.
%C Amicable numbers A063990 are the terms of this sequence in increasing order.
%C First differs from both A063990 and A259180 at a(17).
%H Laszlo Hars, <a href="https://www.mail-archive.com/julia-users@googlegroups.com/msg04022.html">Performance compared to mathematica</a> Julia-users (2014)
%H Khelleos, <a href="http://www.cyberforum.ru/lisp/thread386611.html">Amicable numbers</a>, CyberForum.ru (2011)
%H OEIS Wiki, <a href="https://oeis.org/wiki/Amicable_numbers">Amicable numbers</a> (This page needs work)
%H Wikipédia, <a href="https://hu.wikipedia.org/wiki/Barátságos_számok">Barátságos számok</a> (contains a mistake: A063990 should be replaced with A259933)
%F a(2n-1) + a(2n) = A000203(a(2n-1)) = A000203(a(2n)) = A259953(n).
%e -----------------------------------
%e Amicable pair Sum
%e x y x + y
%e -----------------------------------
%e n A260086 A260087 A259953
%e -----------------------------------
%e 1 220 284 504
%e 2 1184 1210 2394
%e 3 2620 2924 5544
%e 4 5020 5564 10584
%e 5 6232 6368 12600
%e 6 10744 10856 21600
%e 7 12285 14595 26880
%e 8 17296 18416 35712
%e 9 66928 66992 133920
%e 10 67095 71145 138240
%e 11 63020 76084 139104
%e 12 69615 87633 157248
%e ... ... ... ...
%e 32 609928 686072 1296000
%e 33 643336 652664 1296000
%e ...
%e The sum of the proper divisors (or aliquot parts) of 220 is 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284. On the other hand the sum of the proper divisors (or aliquot parts) of 284 is 1 + 2 + 4 + 71 + 142 = 220. Note that 220 + 284 = sigma(220) = sigma(284) = 504. The sum 220 + 284 = 504 is the smallest sum of an amicable pair, so a(1) = 220 and a(2) = 284.
%e Note that some pairs (x, y) share the same sum (x + y), for example: (609928 + 686072) = (643336 + 652664) = sigma(609928) = sigma(686072) = sigma(643336) = sigma(652664) = 1296000, thus in the list first appears the pair (609928, 686072) and then (643336, 652664) because 609928 < 643336.
%Y Cf. A000203, A063990, A259180, A259953, A260086, A260087.
%K nonn
%O 1,1
%A _Omar E. Pol_, Jul 09 2015
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