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A063990
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Amicable numbers.
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38
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220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 10744, 10856, 12285, 14595, 17296, 18416, 63020, 66928, 66992, 67095, 69615, 71145, 76084, 79750, 87633, 88730, 100485, 122265, 122368, 123152, 124155, 139815, 141664, 142310
(list;
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refs;
listen;
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internal format)
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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Scott T. Cohen, Mathematical Buds, Ed. H. D. Ruderman, Vol. 1 Chap. VIII pp. 103-126 Mu Alpha Theta 1984.
C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 90.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 145-7, Penguin Books 1987.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..77977 (terms < 10^14 from Pedersen's tables)
T. Andreescu, Number Theory Trivia: Amicable Numbers
T. Andreescu, Number Theory Trivia: Amicable Numbers
Anonymous, Amicable Pairs Applet Test
Anonymous, Amicable and Social Numbers
G. D'Abramo, On Amicable Numbers With Different Parity
L. Euler, On amicable numbers
M. Garcia, A Million New Amicable Pairs, J. Integer Sequences, 4 (2001), #01.2.6.
M. Garcia, J. M. Pedersen and H. J. J. te Riele, Amicable Pairs, a Survey
Hisanori Mishima, Amicable Numbers:first 236 pairs(smaller member<10^8) fully factorized
D. Moews, A List Of The First 5001 Amicable Pairs
D. and P. C. Moews, A List Of Amicable Pairs Below 2.01*10^11
Number Theory List, NMBRTHRY Archives--August 1993
J. O. M. Pedersen, Known Amicable Pairs
J. O. M. Pedersen, Tables of Aliquot Cycles
I. Peterson, MathTrek, Appealing Numbers
I. Peterson, MathTrek, Amicable Pairs, Divisors and a New Record
H. J. J. te Riele, On Generating New Amicable Pairs from Given Amicable Pairs
H. J. J. te Riele, Computation of All the Amicable Pairs Below 10^10
H. J. J. te Riele, A New Method for Finding Amicable Pairs
E. Sandifer, Amicable numbers
G. Villemin's Almanach of Numbers, Nombres amiables et sociables
Eric Weisstein's World of Mathematics, Amicable Pair
Wikipedia, Amicable number
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MATHEMATICA
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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
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PROG
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Contribution from Michael B. Porter, Apr 13 2010: (Start)
(PARI) aliquot(n)=sigma(n)-n
isA063990(n)={local(a); a=aliquot(n); a<>n && aliquot(a)==n} (End)
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CROSSREFS
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Union of A002025 and A002046.
Sequence in context: A217160 A203777 A121507 * A157107 A175738 A184543
Adjacent sequences: A063987 A063988 A063989 * A063991 A063992 A063993
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, Sep 18 2001
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STATUS
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approved
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