

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
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OFFSET

1,1


COMMENTS

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^(n1))  1, q = 3*(2^n)  1 and r = 9*(2^(2n1))  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


REFERENCES

Scott T. Cohen, Mathematical Buds, Ed. H. D. Ruderman, Vol. 1 Chap. VIII pp. 103126 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. 1457, Penguin Books 1987.


LINKS

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
Leonhard Euler, On amicable numbers
Mariano Garcia, A Million New Amicable Pairs, J. Integer Sequences, 4 (2001), #01.2.6.
Mariano Garcia, Jan Munch Pedersen and H. J. J. te Riele, Amicable Pairs, a Survey
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 ArchivesAugust 1993
Jan Munch Pedersen, Known Amicable Pairs
Jan Munch Pedersen, Tables of Aliquot Cycles
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
Herman J. J. te Riele, Computation of All the Amicable Pairs Below 10^10
Herman J. J. te Riele, A New Method for Finding Amicable Pairs
Ed Sandifer, Amicable numbers
GĂ©rard Villemin's Almanach of Numbers, Nombres amiables et sociables
Eric Weisstein's World of Mathematics, Amicable Pair
Wikipedia, Amicable number


MATHEMATICA

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 *)


PROG

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


CROSSREFS

Union of A002025 and A002046.
Sequence in context: A217160 A203777 A121507 * A233538 A157107 A175738
Adjacent sequences: A063987 A063988 A063989 * A063991 A063992 A063993


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Sep 18 2001


STATUS

approved



