

A063990


Amicable numbers.


117



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
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
Sierpiński (1964), page 176, mentions Erdős's work on the number of pairs of amicable numbers <= x.  N. J. A. Sloane, Dec 27 2017
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


REFERENCES

Scott T. Cohen, Mathematical Buds, Ed. Harry D. Ruderman, Vol. 1, Chap. VIII, pp. 103126, Mu Alpha Theta, 1984.
Clifford A. Pickover, The Math Book, Sterling, NY, 2009; see p. 90.
Wacław Sierpiński, Elementary Theory of Numbers, Panst. Wyd. Nauk, Warsaw, 1964.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 145147.


LINKS

Mariano García, Jan Munch Pedersen, Herman te Riele, Amicable pairs, a survey, Report MASR0307, Centrum Wiskunde & Informatica.


FORMULA

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
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


MAPLE

F:= proc(t) option remember; numtheory:sigma(t)t end proc:
select(t > F(t) <> t and F(F(t))=t, [$1.. 200000]); # Robert Israel, Jun 22 2015


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 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


CROSSREFS

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


KEYWORD

nonn


AUTHOR



STATUS

approved



