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%I #164 Nov 17 2023 11:50:03
%S 220,284,1184,1210,2620,2924,5020,5564,6232,6368,10744,10856,12285,
%T 14595,17296,18416,63020,66928,66992,67095,69615,71145,76084,79750,
%U 87633,88730,100485,122265,122368,123152,124155,139815,141664,142310
%N Amicable numbers.
%C A pair of numbers x and y is called amicable if the sum of the proper divisors of either one is equal to the other. The smallest pair is x = 220, y = 284.
%C The sequence lists the amicable numbers in increasing order. Note that the pairs x, y are not adjacent to each other in the list. See also A002025 for the x's, A002046 for the y's.
%C Theorem: If the three numbers p = 3*(2^(n-1)) - 1, q = 3*(2^n) - 1 and r = 9*(2^(2n-1)) - 1 are all prime where n >= 2, then p*q*(2^n) and r*(2^n) are amicable numbers. This 9th century theorem is due to Thabit ibn Kurrah (see for example, the History of Mathematics by David M. Burton, 6th ed., p. 510). - _Mohammad K. Azarian_, May 19 2008
%C The first time a pair ordered by its first element is not adjacent is x = 63020, y = 76084 which correspond to a(17) and a(23), respectively. - _Omar E. Pol_, Jun 22 2015
%C For amicable pairs see A259180 and also A259933. First differs from A259180 (amicable pairs) at a(18). - _Omar E. Pol_, Jun 01 2017
%C Sierpiński (1964), page 176, mentions Erdős's work on the number of pairs of amicable numbers <= x. - _N. J. A. Sloane_, Dec 27 2017
%C Kanold (1954) proved that the asymptotic upper density of amicable numbers is < 0.204 and Erdős (1955) proved that it is 0. - _Amiram Eldar_, Feb 13 2021
%D Scott T. Cohen, Mathematical Buds, Ed. Harry D. Ruderman, Vol. 1, Chap. VIII, pp. 103-126, Mu Alpha Theta, 1984.
%D Clifford A. Pickover, The Math Book, Sterling, NY, 2009; see p. 90.
%D Wacław Sierpiński, Elementary Theory of Numbers, Panst. Wyd. Nauk, Warsaw, 1964.
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 145-147.
%H T. D. Noe, <a href="/A063990/b063990.txt">Table of n, a(n) for n = 1..77977</a> (terms < 10^14 from Pedersen's tables)
%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 Jonathan Bayless and Dominic Klyve, <a href="http://math.colgate.edu/~integers/a3.5int2009/a3.5int2009.pdf">On the sum of reciprocals of amicable numbers</a>, Integers, Vol. 11A (2011), Article 5.
%H Sergei Chernykh, <a href="/A063990/a063990-6M.zip">Table of n, a(n) for n = 1..823818, zipped file</a> (results of an exhaustive search for all amicable pairs with smaller member < 10^17).
%H Sergei Chernykh, <a href="http://sech.me/ap/">Amicable pairs list</a>.
%H Germano 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 Paul Erdős, <a href="https://users.renyi.hu/~p_erdos/1955-03.pdf">On amicable numbers</a>, Pub. Math. Debrecen, Vol. 4 (1955), pp. 108-111.
%H Leonhard Euler, <a href="http://arXiv.org/abs/math.HO/0409196">On amicable numbers</a>, arXiv:math/0409196 [math.HO], 2004-2009.
%H Steven Finch, <a href="/A000396/a000396.pdf">Amicable Pairs and Aliquot Sequences</a>, 2013. [Cached copy, with permission of the author]
%H Mariano García, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/GARCIA/millionc.html">A Million New Amicable Pairs</a>, J. Integer Sequences, Vol. 4 (2001), Article #01.2.6.
%H Mariano García, Jan Munch Pedersen, Herman 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 Hans-Joachim Kanold, <a href="https://doi.org/10.1007/BF01181341">Über die Dichten der Mengen der vollkommenen und der befreundeten Zahlen</a>, Mathematische Zeitschrift, Vol. 61 (1954), pp. 180-185.
%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 Hanh My Nguyen and Carl Pomerance, <a href="https://doi.org/10.1090/mcom/3362">The reciprocal sum of the amicable numbers</a>, Mathematics of Computation, Vol. 88, No. 317 (2019), pp. 1503-1526, <a href="https://math.dartmouth.edu/~carlp/mcom3362.pdf">alternative link</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 J. O. M. Pedersen, <a href="http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a>. [Broken link]
%H J. O. M. Pedersen, <a href="http://web.archive.org/web/20140502102524/http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a>. [Via Internet Archive Wayback-Machine]
%H J. O. M. Pedersen, <a href="/A063990/a063990.pdf">Tables of Aliquot Cycles</a> [Cached copy, pdf file only]
%H Ivars Peterson, <a href="https://web.archive.org/web/20130628060857/http://www.maa.org/mathland/mathtrek_2_26_01.html">Appealing Numbers</a>, MathTrek, 2001.
%H Ivars Peterson, <a href="https://web.archive.org/web/20130126192043/http://maa.org/mathland/mathtrek_02_02_04.html">Amicable Pairs, Divisors and a New Record</a>, MathTrek, 2004.
%H Paul Pollack, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Pollack/pollack3.html">Quasi-Amicable Numbers are Rare</a>, J. Int. Seq., Vol. 14 (2011), Article # 11.5.2.
%H Carl Pomerance, <a href="https://doi.org/10.1007/978-3-319-22240-0_19">On amicable numbers</a>, in: C. Pomerance and M. Rassias M. (eds.), Analytic number theory, Springer, Cham, 2015, pp. 321-327; <a href="https://math.dartmouth.edu/~carlp/amicablesv3.pdf">alternative link</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., Vol. 42, No. 165 (1984), pp. 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., Vol. 47, No. 175 (1986), pp. 361-368 and Supplement pp. S9-S40.
%H Herman J. J. te Riele, <a href="https://core.ac.uk/download/pdf/301661745.pdf">A New Method for Finding Amicable Pairs</a>, Proceedings of Symposia in Applied Mathematics, Volume 48, 1994.
%H Ed Sandifer, <a href="https://web.archive.org/web/20130126165856/http://maa.org/editorial/euler/How%20Euler%20Did%20It%2025%20amicable%20numbers.pdf">Amicable numbers</a>.
%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 Pomerance shows that there are at most x/exp(sqrt(log x log log log x)/(2 + o(1))) terms up to x for sufficiently large x. - _Charles R Greathouse IV_, Jul 21 2015
%F Sum_{n>=1} 1/a(n) is in the interval (0.0119841556, 215) (Nguyen and Pomerance, 2019; an upper bound 6.56*10^8 was given by Bayless and Klyve, 2011). - _Amiram Eldar_, Oct 15 2020
%p F:= proc(t) option remember; numtheory:-sigma(t)-t end proc:
%p select(t -> F(t) <> t and F(F(t))=t, [$1.. 200000]); # _Robert Israel_, Jun 22 2015
%t s[n_] := DivisorSigma[1, n] - n; AmicableNumberQ[n_] := If[Nest[s, n, 2] == n && ! s[n] == n, True, False]; Select[Range[10^6], AmicableNumberQ[ # ] &] (* _Ant King_, Jan 02 2007 *)
%o (PARI) aliquot(n)=sigma(n)-n
%o isA063990(n)={local(a);a=aliquot(n);a<>n && aliquot(a)==n} \\ _Michael B. Porter_, Apr 13 2010
%o (Python)
%o from sympy import divisors
%o A063990 = [n for n in range(1,10**5) if sum(divisors(n))-2*n and not sum(divisors(sum(divisors(n))-n))-sum(divisors(n))] # _Chai Wah Wu_, Aug 14 2014
%Y Union of A002025 and A002046.
%Y A180164 (gives for each pair (x, y) the value x+y = sigma(x)+sigma(y)).
%Y Cf. A259180.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Sep 18 2001
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