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%I #74 Nov 17 2023 11:51:17
%S 220,284,1184,1210,2620,2924,5020,5564,6232,6368,10744,10856,12285,
%T 14595,17296,18416,63020,76084,66928,66992,67095,71145,69615,87633,
%U 79750,88730,100485,124155,122265,139815,122368,123152,141664,153176,142310,168730,171856,176336,176272,180848,185368,203432,196724,202444,280540,365084
%N Amicable pairs.
%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 This is A002025 and A002046 interleaved hence the amicable pairs (x < y), ordered by increasing x, are adjacent to each other in the list.
%C By definition a property of the amicable pair (x, y) is that x + y = sigma(x) = sigma(y).
%C Amicable numbers A063990 are the terms of this sequence in increasing order.
%C First differs from A063990 at a(18).
%C For another version see A259933.
%C First differs from A259933 at a(17).
%H Titu Andreescu, <a href="http://staff.imsa.edu/math/journal/volume3/articles/NumberTrivia.pdf">Number Theory Trivia: Amicable Numbers</a>
%H Titu Andreescu, <a href="http://britton.disted.camosun.bc.ca/amicable.html">Number Theory Trivia: Amicable Numbers</a>
%H Anonymous, <a href="http://nautilus.fis.uc.pt/mn/i_amigos/amigos.swf">Amicable Pairs Applet Test</a>
%H Anonymous, <a href="http://www-maths.swan.ac.uk/pgrads/bb/project/node16.html">Amicable and Social Numbers</a> [broken link]
%H S. Chernykh, <a href="https://sech.me/boinc/Amicable/">Amicable Numbers</a>
%H S. Chernykh, <a href="http://sech.me/ap/">Amicable pairs list</a>
%H G. D'Abramo, <a href="http://arXiv.org/abs/math.HO/0501402">On Amicable Numbers With Different Parity</a>, arXiv:math/0501402 [math.HO], 2005-2007.
%H E. B. Escott, <a href="/A002025/a002025.pdf">Amicable numbers</a>, Scripta Mathematica, 12 (1946), 61-72 [Annotated scanned copy]
%H Leonhard Euler, <a href="http://arXiv.org/abs/math.HO/0409196">On amicable numbers</a>, arXiv:math/0409196 [math.HO], 2004-2009.
%H Mariano Garcia, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/GARCIA/millionc.html">A Million New Amicable Pairs</a>, J. Integer Sequences, 4 (2001), #01.2.6.
%H M. García, J. M. Pedersen, H. J. J. te Riele, <a href="http://oai.cwi.nl/oai/asset/4143/04143D.pdf">Amicable pairs, a survey</a>, Report MAS-R0307, Centrum Wiskunde & Informatica.
%H S. S. Gupta, <a href="http://www.shyamsundergupta.com/amicable.htm">Amicable Numbers</a>
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~kc2h-msm/mathland/math09/ami02.htm">Amicable Numbers:first 236 pairs(smaller member<10^8) fully factorized</a>
%H David Moews, <a href="http://djm.cc/amicable2.txt">A List Of The First 5001 Amicable Pairs</a>
%H David and P. C. Moews, <a href="http://djm.cc/amicable.txt">A List Of Amicable Pairs Below 2.01*10^11</a>
%H Passawan Noppakaew and Prapanpong Pongsriiam, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Pongsriiam/pong43.html">Product of Some Polynomials and Arithmetic Functions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
%H Number Theory List, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A1=ind9308&L=nmbrthry">NMBRTHRY Archives--August 1993</a>
%H Jan Munch Pedersen, <a href="http://amicable.homepage.dk/knwnap.htm">Known Amicable Pairs</a> [Broken link]
%H Jan Munch Pedersen, <a href="http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a> [Broken link]
%H Ivars Peterson, MathTrek, <a href="http://www.maa.org/mathland/mathtrek_2_26_01.html">Appealing Numbers</a>
%H Ivars Peterson, MathTrek, <a href="http://www.maa.org/mathland/mathtrek_02_02_04.html">Amicable Pairs, Divisors and a New Record</a>
%H Herman J. J. te Riele, <a href="http://dx.doi.org/10.1090/S0025-5718-1984-0725997-0">On generating new amicable pairs from given amicable pairs</a>, Math. Comp. 42 (1984), 219-223.
%H Herman J. J. te Riele, <a href="http://dx.doi.org/10.1090/S0025-5718-1986-0842142-3">Computation of all the amicable pairs below 10^10</a>, Math. Comp., 47 (1986), 361-368 and Supplement pp. S9-S40.
%H Herman J. J. te Riele, <a href="http://oai.cwi.nl/oai/asset/2246/2246A.pdf">A New Method for Finding Amicable Pairs</a>, Proceedings of Symposia in Applied Mathematics, Volume 48, 1994.
%H Ed Sandifer, <a href="http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2025%20amicable%20numbers.pdf">Amicable numbers</a>
%H Juan Luis Varona, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Varona/varona4.html">On the Solution of the Equation n = a*k + b*p_k by Means of an Iterative Method</a>, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.5.
%H Gérard Villemin's Almanach of Numbers, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Decompos/Amiable.htm">Nombres amiables et sociables</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AmicablePair.html">Amicable Pair</a>
%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Amicable_number">Amicable number</a>
%F a(2n-1) = A002025(n); a(2n) = A002046(n).
%F a(2n-1) + a(2n) = A000203(a(2n-1)) = A000203(a(2n)) = A180164(n).
%e ------------------------------------
%e Amicable pair Sum
%e x y x + y
%e ------------------------------------
%e n A002025 A002046 A180164
%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 63020 76084 139104
%e 10 66928 66992 133920
%e 11 67095 71145 138240
%e 12 69615 87633 157248
%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 smallest amicable pair is (220, 284), so a(1) = 220 and a(2) = 284.
%t f[n_] := Block[{s = {}, g, k}, g[x_] := DivisorSigma[1, x] - x; Do[k = g@ i; If[And[g@ k == i, k != i, ! MemberQ[s, i]], s = s~Join~{i, k}], {i, n}]; s]; f@ 300000 (* _Michael De Vlieger_, Jul 02 2015 *)
%o (PARI) A259180_upto(N, L=List(), s)={ forfactored(n=1, N, (s=sigma(n[2]))>2*n[1] && sigma(s-n[1])==s && listput(L, [n[1], s-n[1]]));concat(L)} \\ _M. F. Hasler_, Oct 11 2019
%Y Cf. A000203, A001065, A002025, A002046, A063990, A066539, A180164, A180202, A259933.
%K nonn
%O 1,1
%A _Omar E. Pol_, Jun 20 2015
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