A063990 Amicable numbers.

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

Offset: 1

Author

N. J. A. Sloane, Sep 18 2001

Keywords

• nonn

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^(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 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

For amicable pairs see A259180 and also A259933. First differs from A259180 (amicable pairs) at a(18). - Omar E. Pol, Jun 01 2017

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. 103-126, 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.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 137-141.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 145-147.

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. [broken link]
• Jonathan Bayless and Dominic Klyve, On the sum of reciprocals of amicable numbers, Integers, Vol. 11A (2011), Article 5.
• Sergei Chernykh, Table of n, a(n) for n = 1..823818, zipped file (results of an exhaustive search for all amicable pairs with smaller member < 10^17).
• Sergei Chernykh, Amicable pairs list.
• Germano D'Abramo, On Amicable Numbers With Different Parity, arXiv:math/0501402 [math.HO], 2005-2007.
• Paul Erdős, On amicable numbers, Pub. Math. Debrecen, Vol. 4 (1955), pp. 108-111.
• Leonhard Euler, On amicable numbers, arXiv:math/0409196 [math.HO], 2004-2009.
• Steven Finch, Amicable Pairs and Aliquot Sequences, 2013. [Cached copy, with permission of the author]
• Mariano García, A Million New Amicable Pairs, J. Integer Sequences, Vol. 4 (2001), Article #01.2.6.
• Mariano García, Jan Munch Pedersen, Herman te Riele, Amicable pairs, a survey, Report MAS-R0307, Centrum Wiskunde & Informatica.
• Hans-Joachim Kanold, Über die Dichten der Mengen der vollkommenen und der befreundeten Zahlen, Mathematische Zeitschrift, Vol. 61 (1954), pp. 180-185.
• 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
• Hanh My Nguyen and Carl Pomerance, The reciprocal sum of the amicable numbers, Mathematics of Computation, Vol. 88, No. 317 (2019), pp. 1503-1526, alternative link.
• Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
• Number Theory List, NMBRTHRY Archives--August 1993.
• J. O. M. Pedersen, Tables of Aliquot Cycles. [Broken link]
• J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
• J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]
• Ivars Peterson, Appealing Numbers, MathTrek, 2001.
• Ivars Peterson, Amicable Pairs, Divisors and a New Record, MathTrek, 2004.
• Paul Pollack, Quasi-Amicable Numbers are Rare, J. Int. Seq., Vol. 14 (2011), Article # 11.5.2.
• Carl Pomerance, On amicable numbers, in: C. Pomerance and M. Rassias M. (eds.), Analytic number theory, Springer, Cham, 2015, pp. 321-327; alternative link.
• Herman J. J. te Riele, On generating new amicable pairs from given amicable pairs, Math. Comp., Vol. 42, No. 165 (1984), pp. 219-223.
• Herman J. J. te Riele, Computation of all the amicable pairs below 10^10, Math. Comp., Vol. 47, No. 175 (1986), pp. 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.

Crossrefs

Union of A002025 and A002046.

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

Cf. A259180.

Programs

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 *)
Select[Tally[Sort/@Table[{n,DivisorSigma[1,n]-n},{n,200000}]],#[[2]]==2&][[;;,1]]//Flatten//Sort (* Harvey P. Dale, Jan 13 2025 *)

PARI

aliquot(n)=sigma(n)-n
isA063990(n)={if (n>1, 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

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