%I A063990
%S A063990 220,284,1184,1210,2620,2924,5020,5564,6232,6368,10744,10856,12285,
%T A063990 14595,17296,18416,63020,66928,66992,67095,69615,71145,76084,79750,
%U A063990 87633,88730,100485,122265,122368,123152,124155,139815,141664,142310
%N A063990 Amicable numbers.
%C A063990 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 (azarian(AT)evansville.edu), May 19 
               2008
%C A063990 Conjecture:Let p = prime number; if 2^n=[2*(p+1)/3]; b=2*p+1 (with b=prime); 
               c=2*p^2+4*p+1 (with c=prime); then p*b*2^n and c*2^n are amicable 
               numbers. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Aug 
               15 2009]
%D A063990 Scott T. Cohen, Mathematical Buds, Ed. H. D. Ruderman, Vol. 1 Chap. VIII 
               pp. 103-126 Mu Alpha Theta 1984.
%D A063990 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, 
               pp. 145-7, Penguin Books 1987.
%D A063990 C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 90.
%H A063990 T. D. Noe, <a href="b063990.txt">Amicable numbers less than 10^12, table 
               of n, a(n) for n=1..15109</a> (from Pedersen's tables)
%H A063990 T. Andreescu, <a href="http://staff.imsa.edu/math/journal/volume3/articles/
               NumberTrivia.pdf">Number Theory Trivia: Amicable Numbers</a>
%H A063990 T. Andreescu, <a href="http://britton.disted.camosun.bc.ca/amicable.html">
               Number Theory Trivia: Amicable Numbers</a>
%H A063990 Anonymous, <a href="http://nautilus.fis.uc.pt/mn/i_amigos/amigos.swf">
               Amicable Pairs Applet Test</a>
%H A063990 Anonymous, <a href="http://www-maths.swan.ac.uk/pgrads/bb/project/node16.html">
               Amicable and Social Numbers</a>
%H A063990 G. D'Abramo, <a href="http://arXiv.org/abs/math.HO/0501402">On Amicable 
               Numbers With Different Parity</a>
%H A063990 L. Euler, <a href="http://arXiv.org/abs/math.HO/0409196">On amicable 
               numbers</a>
%H A063990 M. 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 A063990 M. Garcia, J. M. Pedersen and H. J. J. te Riele, <a href="http://db.cwi.nl/
               rapporten/abstract.php?abstractnr=1368">Amicable Pairs, a Survey</
               a>
%H A063990 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 A063990 D. Moews, <a href="http://xraysgi.ims.uconn.edu/amicable2.txt">A List 
               Of The First 5001 Amicable Pairs</a>
%H A063990 D. and P. C. Moews, <a href="http://djm.cc/amicable.txt">A List Of Amicable 
               Pairs Below 2.01*10^11</a>
%H A063990 J. O. M. Pedersen, <a href="http://amicable.homepage.dk/knwnap.htm">Known 
               Amicable Pairs</a>
%H A063990 J. O. M. Pedersen, <a href="http://amicable.homepage.dk/tables.htm">Tables 
               of Aliquot Cycles</a>
%H A063990 I. Peterson, MathTrek, <a href="http://www.maa.org/mathland/mathtrek_2_26_01.html">
               Appealing Numbers</a>
%H A063990 I. Peterson, MathTrek, <a href="http://www.maa.org/mathland/mathtrek_02_02_04.html">
               Amicable Pairs, Divisors and a New Record</a>
%H A063990 H. J. J. te Riele, <a href="http://repos.project.cwi.nl:8888/cwi_repository/
               docs/I/02/2587A.pdf">On Generating New Amicable Pairs from Given 
               Amicable Pairs</a>
%H A063990 H. J. J. te Riele, <a href="http://repos.project.cwi.nl:8888/cwi_repository/
               docs/I/06/6396A.pdf">Computation of All the Amicable Pairs Below 
               10^10</a>
%H A063990 H. J. J. te Riele, <a href="http://db.cwi.nl/rapporten/abstract.php?abstractnr=884">
               A New Method for Finding Amicable Pairs</a>
%H A063990 E. Sandifer, <a href="http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2025%20amicable%20numbers.pdf\
               ">Amicable numbers</a>
%H A063990 G. Villemin's Almanach of Numbers, <a href="http://membres.lycos.fr/villemingerard/
               Decompos/Amiable.htm">Nombres amiables et sociables</a>
%H A063990 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               AmicablePair.html">Link to a section of The World of Mathematics</
               a>
%H A063990 Wikipedia, <a href="http://www.wikipedia.org/wiki/Amicable_number">Amicable 
               number</a>
%H A063990 Number Theory List, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A1=ind9308&L=nmbrthry">
               NMBRTHRY Archives--August 1993</a>
%t A063990 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 (mathstutoring(AT)ntlworld.com), Jan 02 2007
%Y A063990 Union of A002025 and A002046.
%Y A063990 Sequence in context: A157673 A064477 A121507 this_sequence A157107 A135807 
               A102073
%Y A063990 Adjacent sequences: A063987 A063988 A063989 this_sequence A063991 A063992 
               A063993
%K A063990 nonn
%O A063990 1,1
%A A063990 N. J. A. Sloane (njas(AT)research.att.com), Sep 18 2001