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A063990 Amicable numbers. 42
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; graph; refs; listen; history; text; internal format)



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.

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.

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

The first time a pair is not adjacent is x = 67095, y = 71145 which correspond to a(20) and a(22), respectively. - Jeppe Stig Nielsen, Jan 27 2015


Scott T. Cohen, Mathematical Buds, Ed. H. D. Ruderman, Vol. 1 Chap. VIII pp. 103-126 Mu Alpha Theta 1984.

Clifford A. Pickover, The Math Book, Sterling, NY, 2009; see p. 90.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 145-7, Penguin Books 1987.


T. D. Noe, Table of n, a(n) for n=1..77977 (terms < 10^14 from Pedersen's tables)

Titu Andreescu, Number Theory Trivia: Amicable Numbers

Titu 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, arXiv:math/0501402 [math.HO], 2005-2007.

Leonhard Euler, On amicable numbers, arXiv:math/0409196 [math.HO], 2004-2009.

Mariano Garcia, A Million New Amicable Pairs, J. Integer Sequences, 4 (2001), #01.2.6.

M. García, J. M. Pedersen, H. J. J. te Riele, Amicable pairs, a survey, Report MAS-R0307, Centrum Wiskunde & Informatica.

Hisanori Mishima, Amicable Numbers:first 236 pairs(smaller member<10^8) fully factorized

David Moews, A List Of The First 5001 Amicable Pairs

David and P. C. Moews, A List Of Amicable Pairs Below 2.01*10^11

Number Theory List, NMBRTHRY Archives--August 1993

Jan Munch Pedersen, Known Amicable Pairs [Broken link]

Jan Munch Pedersen, Tables of Aliquot Cycles [Broken link]

Ivars Peterson, MathTrek, Appealing Numbers

Ivars Peterson, MathTrek, Amicable Pairs, Divisors and a New Record

Herman J. J. te Riele, On generating new amicable pairs from given amicable pairs, Math. Comp. 42 (1984), 219-223.

Herman J. J. te Riele, Computation of all the amicable pairs below 10^10, Math. Comp., 47 (1986), 361-368 and Supplement pp. S9-S40.

Herman J. J. te Riele, A New Method for Finding Amicable Pairs, Proceedings of Symposia in Applied Mathematics, Volume 48, 1994.

Ed Sandifer, Amicable numbers

Gérard Villemin's Almanach of Numbers, Nombres amiables et sociables

Eric Weisstein's World of Mathematics, Amicable Pair

Wikipedia, Amicable number


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 *)


(PARI) aliquot(n)=sigma(n)-n

isA063990(n)={local(a); a=aliquot(n); a<>n && aliquot(a)==n} \\ Michael B. Porter, Apr 13 2010


from sympy import divisors

A063990 = [n for n in xrange(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


Union of A002025 and A002046.

A180164 (gives for each pair (x, y) the value x+y = sigma(x)+sigma(y)).

Sequence in context: A217160 A203777 A121507 * A233538 A157107 A175738

Adjacent sequences:  A063987 A063988 A063989 * A063991 A063992 A063993




N. J. A. Sloane, Sep 18 2001



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Last modified March 4 19:23 EST 2015. Contains 255187 sequences.