A259180 -id:A259180 - cp's OEIS frontend

A328009 Irregular array read by rows in which row n lists the divisors of the n-th term of the sequence of amicable pairs (A259180).

1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220, 1, 2, 4, 71, 142, 284, 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592, 1184, 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605, 1210, 1, 2, 4, 5, 10, 20, 131, 262, 524, 655, 1310, 2620, 1, 2, 4, 17, 34, 43, 68, 86, 172, 731, 1462, 2924, 1, 2, 4, 5, 10, 20, 251, 502, 1004, 1255

Offset: 1

Omar E. Pol, Oct 01 2019

Row sums give a sequence formed by the terms of A180164 repeated as follows: 504, 504, 2394, 2394, 5544, 5544, ...

			Array begins:
1, 2, 4,  5,  10,  11,  20,  22,   44,  55,   110,  220;
1, 2, 4, 71, 142, 284;
1, 2, 4,  8,  16,  32,  37,  74,  148,  296,  592, 1184;
1, 2, 5, 10,  11,  22,  55, 110,  121,  242,  605, 1210;
1, 2, 4,  5,  10,  20, 131, 262,  524,  655, 1310, 2620;
1, 2, 4, 17,  34,  43,  68,  86,  172,  731, 1462, 2924;
1, 2, 4,  5,  10,  20, 251, 502, 1004, 1255, 2510, 5020;
1, 2, 4, 13,  26,  52, 107, 214,  428, 1391, 2782, 5564,
1, 2, 4,  8,  19,  38,  41,  76,   82,  152,  164,  328, 779, 1558, 3116, 6232;
1, 2, 4,  8,  16,  32, 199, 398,  796, 1592, 3184, 6368;
...

Right border gives A259180 (amicable pairs).
The length of row n is A328043(n).
Column 1 gives A000012.
Cf. A002025, A002046, A018339, A018373, A027750, A180164, A328063, A328064,  A328065.

With[{s = Array[{#, DivisorSigma[1, #] - #} &, 6000]}, Flatten@ Divisors@ DeleteDuplicates[Sort /@ Select[Reverse /@ s, And[! FreeQ[s, #], UnsameQ @@ #] &]]] (* Michael De Vlieger, Oct 08 2019 *)

A001065 Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 36, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 55, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 8, 43, 21, 46, 1, 66, 17, 64, 23, 32, 1, 108, 1, 34, 41, 63, 19, 78, 1, 58, 27, 74, 1, 123, 1, 40, 49, 64, 19, 90, 1, 106

Offset: 1

N. J. A. Sloane, R. K. Guy

Also total number of parts in all partitions of n into equal parts that do not contain 1 as a part. - Omar E. Pol, Jan 16 2013
Related concepts: If a(n) < n, n is said to be deficient, if a(n) > n, n is abundant, and if a(n) = n, n is perfect. If there is a cycle of length 2, so that a(n) = b and a(b) = n, b and n are said to be amicable. If there is a longer cycle, the numbers in the cycle are said to be sociable. See examples. - Juhani Heino, Jul 17 2017
Sum of the smallest parts in the partitions of n into two parts such that the smallest part divides the largest. - Wesley Ivan Hurt, Dec 22 2017
a(n) is also the total number of parts congruent to 0 mod k in the partitions of k*n into equal parts that do not contain k as a part (the comment dated Jan 16 2013 is the case for k = 1). - Omar E. Pol, Nov 23 2019
Fixed points are in A000396. - Alois P. Heinz, Mar 10 2024

			x^2 + x^3 + 3*x^4 + x^5 + 6*x^6 + x^7 + 7*x^8 + 4*x^9 + 8*x^10 + x^11 + ...
For n = 44, sum of divisors of n = sigma(n) = 84; so a(44) = 84-44 = 40.
Related concepts: (Start)
From 1 to 17, all n are deficient, except 6 and 12 seen below. See A005100.
Abundant numbers: a(12) = 16, a(18) = 21. See A005101.
Perfect numbers: a(6) = 6, a(28) = 28. See A000396.
Amicable numbers: a(220) = 284, a(284) = 220. See A259180.
Sociable numbers: 12496 -> 14288 -> 15472 -> 14536 -> 14264 -> 12496. See A122726. (End)
For n = 10 the sum of the divisors of 10 that are less than 10 is 1 + 2 + 5 = 8. On the other hand, the partitions of 10 into equal parts that do not contain 1 as a part are [10], [5,5], [2,2,2,2,2], there are 8 parts, so a(10) = 8. - _Omar E. Pol_, Nov 24 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
George E. Andrews, Number Theory. New York: Dover, 1994; Pages 1, 75-92; p. 92 #15: Sigma(n) / d(n) >= n^(1/2).
Carl Pomerance, The first function and its iterates, pp. 125-138 in Connections in Discrete Mathematics, ed. S. Butler et al., Cambridge, 2018.
H. J. J. te Riele, Perfect numbers and aliquot sequences, pp. 77-94 in J. van de Lune, ed., Studieweek "Getaltheorie en Computers", published by Math. Centrum, Amsterdam, Sept. 1980.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 91.

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972. [alternative scanned copy].
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Henry Bottomley, Illustration of initial terms
K. Chum, R. K. Guy, M. J. Jacobson, Jr., and A. S. Mosunov, Numerical and statistical analysis of aliquot sequences, Exper. Math. 29 (2020), no. 4, 414-425; arXiv:2110.14136, Oct. 2021 [math.NT].
Don Coppersmith, An answer to the problem of Don Saari, 1987.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
P. Pollack and C. Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B 3 (2016), 1-26; errata.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp., to appear (2014).
Primefan, Sums of Restricted Divisors for n=1 to 1000
F. Richman, Aliquot series: Abundant, deficient, perfect
Eric Weisstein's World of Mathematics, Restricted Divisor Function
Eric Weisstein's World of Mathematics, Divisor Function
Index entries for "core" sequences

Least inverse: A070015, A359132.
Values taken: A078923, values not taken: A005114.
Records: A034090, A034091.
First differences: A053246, partial sums: A153485.
a(n) = n - A033879(n) = n + A033880(n). - Omar E. Pol, Dec 30 2013
Row sums of A141846 and of A176891. - Gary W. Adamson, May 02 2010
Row sums of A176079. - Mats Granvik, May 20 2012
Alternating row sums of A231347. - Omar E. Pol, Jan 02 2014
a(n) = sum (A027751(n,k): k = 1..A000005(n)-1). - Reinhard Zumkeller, Apr 05 2013
For n > 1: a(n) = A240698(n,A000005(n)-1). - Reinhard Zumkeller, Apr 10 2014
A134675(n) = A007434(n) + a(n). - Conjectured by John Mason and proved by Max Alekseyev, Jan 07 2015
Cf. A032741, A000203, A048050, A000593, A027750.
Cf. A051953, A051731.
Cf. A037020 (primes), A053868, A053869 (odd and even terms).
Cf. A048138 (number of occurrences), A238895, A238896 (record values thereof).
Cf. A007956 (products of proper divisors).
Cf. A005100, A005101, A000396, A259180, A122726 (related concepts).

a001065 n = a000203 n - n  -- Reinhard Zumkeller, Sep 15 2011

[SumOfDivisors(n)-n: n in [1..100]]; // Vincenzo Librandi, May 06 2015

A001065 := proc(n)
    numtheory[sigma](n)-n ;
end proc:
seq( A001065(n),n=1..100) ;

Table[ Plus @@ Select[ Divisors[ n ], #Zak Seidov, Sep 10 2009 *)
Table[DivisorSigma[1, n] - n, {n, 1, 80}] (* Jean-François Alcover, Apr 25 2013 *)
Array[Plus @@ Most@ Divisors@# &, 80] (* Robert G. Wilson v, Dec 24 2017 *)

numlib::sigma(n)-n$ n=1..81 // Zerinvary Lajos, May 13 2008

{a(n) = if( n==0, 0, sigma(n) - n)} /* Michael Somos, Sep 20 2011 */

from sympy import divisor_sigma
def A001065(n): return divisor_sigma(n)-n # Chai Wah Wu, Nov 04 2022

[sigma(n, 1)-n for n in range(1, 81)] # Stefano Spezia, Jul 14 2025

G.f.: Sum_{k>0} k * x^(2*k)/(1 - x^k). - Michael Somos, Jul 05 2006
a(n) = sigma(n) - n = A000203(n) - n. - Lekraj Beedassy, Jun 02 2005
a(n) = A155085(-n). - Michael Somos, Sep 20 2011
Equals inverse Mobius transform of A051953 = A051731 * A051953. Example: a(6) = 6 = (1, 1, 1, 0, 0, 1) dot (0, 1, 1, 2, 1, 4) = (0 + 1 + 1 + 0 + 0 + 4), where A051953 = (0, 1, 1, 2, 1, 4, 1, 4, 3, 6, 1, 8, ...) and (1, 1, 1, 0, 0, 1) = row 6 of A051731 where the 1's positions indicate the factors of 6. - Gary W. Adamson, Jul 11 2008
a(n) = A006128(n) - A220477(n) - n. - Omar E. Pol Jan 17 2013
a(n) = Sum_{i=1..floor(n/2)} i*(1-ceiling(frac(n/i))). - Wesley Ivan Hurt, Oct 25 2013
Dirichlet g.f.: zeta(s-1)*(zeta(s) - 1). - Ilya Gutkovskiy, Aug 07 2016
a(n) = 1 + A048050(n), n > 1. - R. J. Mathar, Mar 13 2018
Erdős (Elem. Math. 28 (1973), 83-86) shows that the density of even integers in the range of a(n) is strictly less than 1/2. The argument of Coppersmith (1987) shows that the range of a(n) has density at most 47/48 < 1. - N. J. A. Sloane, Dec 21 2019
G.f.: Sum_{k >= 2} x^k/(1 - x^k)^2. Cf. A296955. (This follows from the fact that if g(z) = Sum_{n >= 1} a(n)*z^n and f(z) = Sum_{n >= 1} a(n)*z^(N*n)/(1 - z^n) then f(z) = Sum_{k >= N} g(z^k), taking a(n) = n and N = 2.) - Peter Bala, Jan 13 2021
Faster converging g.f.: Sum_{n >= 1} q^(n*(n+1))*(n*q^(3*n+2) - (n + 1)*q^(2*n+1) - (n - 1)*q^(n+1) + n)/((1 - q^n)*(1 - q^(n+1))^2). (In equation 1 in Arndt, after combining the two n = 0 summands to get -t/(1 - t), apply the operator t*d/dt to the resulting equation and then set t = q and x = 1.) - Peter Bala, Jan 22 2021
a(n) = Sum_{d|n} d * (1 - [n = d]), where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 28 2021
a(n) = Sum_{i=1..n} ((n-1) mod i) - (n mod i). [See also A176079.] - José de Jesús Camacho Medina, Feb 23 2021

A063990 Amicable numbers.

Original entry on oeis.org

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

N. J. A. Sloane, Sep 18 2001

nonn

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

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.

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.

Union of A002025 and A002046.
A180164 (gives for each pair (x, y) the value x+y = sigma(x)+sigma(y)).
Cf. A259180.

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

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 *)

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

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

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

A002025 Smaller of an amicable pair: (a,b) such that sigma(a) = sigma(b) = a+b, a < b.

Original entry on oeis.org

220, 1184, 2620, 5020, 6232, 10744, 12285, 17296, 63020, 66928, 67095, 69615, 79750, 100485, 122265, 122368, 141664, 142310, 171856, 176272, 185368, 196724, 280540, 308620, 319550, 356408, 437456, 469028, 503056, 522405, 600392, 609928

Offset: 1

N. J. A. Sloane

Sometimes called friendly numbers, but this usage is deprecated.
All terms are abundant (A005101). - Michel Marcus, Mar 10 2013
See A125490-A125492 and A137231 for amicable triples, A036471-A036474 and A116148 for amicable quadruples, and A233553 for amicable quintuples. - M. F. Hasler, Dec 14 2013
This sequence is strictly increasing (and A002046, which contains the larger (deficient) number in each pair, is sorted by this sequence). - Jeppe Stig Nielsen, Jan 27 2015
For the related amicable pairs see A259180. - Omar E. Pol, Jul 15 2015
Pomerance (1981) shows that there are at most x*exp(-log(x)^(1/3)) terms of this sequence up to x. In particular, as originally demonstrated by Erdős, this sequence has density 0. - Charles R Greathouse IV, Aug 17 2017

Mariano Garcia, Jan Munch Pedersen and Herman te Riele, Amicable pairs - a survey, pp. 179-196 in: Alf van der Poorten and Andres Stein (eds.), High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, Fields Institute Communications, AMS, Providence RI, 2004.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 48-49.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Sergei Chernykh, Table of n, a(n) for n = 1..415523 [All terms up to 10^17. Terms 39375 through 415523 were computed by Sergei Chernykh]
J. Alanen, O. Ore and J. Stemple, Systematic computations on amicable numbers, Math. Comp., 21 (1967), 242-245.
J. Bell, A translation of Leonhard Euler's..., arXiv:math/0409196 [math.HO], 2004-2009.
W. Borho and H. Hoffmann, Breeding Amicable Numbers in Abundance, Math. Comp., 46 (1986), 281-293.
S. Chernykh, Amicable pairs list.
Paul Erdős, On amicable numbers, Publ. Math. Debrecen 4 (1955), pp. 108-111.
E. B. Escott, Amicable numbers, Scripta Mathematica, 12 (1946), 61-72. [Annotated scanned copy]
L. Euler, De numeris amicabilibus, Opuscula varii argumetii, pages 23-107, 1750. Reprinted in Opera mathematica: Series prima. Volumen II, Leonhardi Euleri commentationes arithmeticae. Sub ausp. soc. scient. nat. Helv., Teubner, Leipzig, Series I, Vol. 1915, pp. 86-162.
M. Garcia, A Million New Amicable Pairs, J. Integer Seqs., Vol. 4 (2001), #01.2.6.
Mariano García, Jan Munch Pedersen, and Herman J. J. te Riele, Amicable pairs, a survey, Report MAS-R0307, 2003, Centrum Wiskunde en Informatica.
Mariano García, Jan Munch Pedersen, and Herman J. J. te Riele, Amicable pairs, a survey, Fields Institute Comm. (2004) Vol. 41.
S. S. Gupta, Amicable Numbers.
E. J. Lee, Amicable Numbers and the Bilinear Diophantine Equation, Math. Comp., 22 (1968), 181-187.
Hisanori Mishima, First 236 amicable pairs.
D. Moews, Perfect, amicable and sociable numbers.
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
J. O. M. Pedersen, Known Amicable Pairs. [Broken link]
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]
Carl Pomerance, On the distribution of amicable numbers, J. reine angew. Math. 293/294 (1977), pp. 217-222.
Carl Pomerance, On the distribution of amicable numbers, II, J. reine angew. Math. 325 (1981), pp. 183-188.
H. J. J. te Riele, Four large amicable pairs, Math. Comp., 28 (1974), 309-312.
H. J. J. te Riele, Computation of all the amicable pairs below 10^10, Math. Comp., 47 (1986), 361-368 and Supplement pp. S9-S40.
H. J. J. te Riele et al., Table of Amicable Pairs between 10^10 and 10^52, Note NM-N8603, Department of Numerical Mathematics, Centre for Mathematics and Computer Science, Amsterdam, 1986. (Warning: file size is 65MB.)
T. Trotter, Jr., Amicable Numbers, archived from the original.
Eric Weisstein's World of Mathematics, Amicable Pair.

Cf. A000203, A002046, A063990, A066873, A180164, A259180.

Reap[For[n = 1, n <= 10^6, n++, If[(s = DivisorSigma[1, n]) > 2n && DivisorSigma[1, s - n] == s, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 09 2015, after M. F. Hasler *)

aliquot(n)=sigma(n)-n
isA002025(n)={if (n>1, local(a);a=aliquot(n);a>n && aliquot(a)==n)} \\ Michael B. Porter, Apr 11 2010

for(n=1,1e6,(s=sigma(n))>2*n && sigma(s-n)==s && print1(n",")) \\ M. F. Hasler, Dec 14 2013

forfactored(n=1,10^6, t=sigma(n[2])-n[1]; if(t>n[1] && sigma(t)==n[1]+t, print1(n[1]", "))) \\ Charles R Greathouse IV, Aug 17 2017

a(n) = A259180(2n-1) = A180164(n) - A259180(2n) = A180164(n) - A002046(n). - Omar E. Pol, Jul 15 2015

More terms from Larry Reeves (larryr(AT)acm.org), Oct 24 2000

A002046 Larger of amicable pair.

Original entry on oeis.org

284, 1210, 2924, 5564, 6368, 10856, 14595, 18416, 76084, 66992, 71145, 87633, 88730, 124155, 139815, 123152, 153176, 168730, 176336, 180848, 203432, 202444, 365084, 389924, 430402, 399592, 455344, 486178, 514736, 525915, 669688, 686072

Offset: 1

N. J. A. Sloane

The elements 76084, 123152, etc. are intentionally out of numerical order so that a(n) and A002025(n) form amicable pairs. - Michael B. Porter, Apr 17 2010
All terms are deficient (A005100). - Michel Marcus, Mar 10 2013
For the related amicable pairs see A259180. - Omar E. Pol, Jul 15 2015

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
For additional references see A002025.

T. D. Noe and Sergei Chernykh, Table of n, a(n) for n = 1..415523 [All terms such that the smaller number A002025(n) is < 10^17. Terms 39375 through 415523 were computed by Sergei Chernykh]
J. Bell, A translation of Leonhard Euler's "On amicable numbers", arXiv:math/0409196 [math.HO], 2004-2009.
S. Chernykh, Amicable pairs list
E. B. Escott, Amicable numbers, Scripta Mathematica, 12 (1946), 61-72 [Annotated scanned copy]
M. Garcia, A Million New Amicable Pairs, J. Integer Seqs., Vol. 4 (2001), #01.2.6.
S. S. Gupta, Amicable Numbers
Hisanori Mishima, First 236 amicable pairs
David Moews, Perfect, amicable and sociable numbers
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
J. O. M. Pedersen, Known Amicable Pairs [Broken link]
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]
T. Trotter, Jr., Amicable Numbers
Eric Weisstein's World of Mathematics, Amicable Pair

Cf. A002025, A063990, A180164, A259180.

f:= proc(t) uses numtheory; local s;
  s:= sigma(t) - t; s > t and sigma(s) - s = t
end proc;
Am1:= select(f,[$1..10^6]);
map(numtheory:-sigma,Am1); # Robert Israel, Jul 16 2015

amicableQ[n_] := With[{s = DivisorSigma[1, n] - n}, r = n != s && n == DivisorSigma[1, s] - s; If[r, mate[n] = s; True, False]]; mate /@ Select[ Range[lim], amicableQ[#] && # < mate[#] &] (* Jean-François Alcover, Sep 20 2011 *)
Table[DivisorSigma[1, A002025[n]] - A002025[n], {n, 50}] (* T. D. Noe, Sep 20 2011 *)

aliquot(n)=sigma(n)-n
isA002046(n)={if (n>1, local(a);a=aliquot(n);aMichael B. Porter, Apr 17 2010

a(n) = A259180(2n) = A180164(n) - A259180(2n-1) = A180164(n) - A002025(n). - Omar E. Pol, Jul 15 2015

More terms from Larry Reeves (larryr(AT)acm.org), Oct 25 2000

A259933 Amicable pairs (x < y) ordered by nondecreasing sum (x + y) and then by increasing x.

Original entry on oeis.org

220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 10744, 10856, 12285, 14595, 17296, 18416, 66928, 66992, 67095, 71145, 63020, 76084, 69615, 87633, 79750, 88730, 100485, 124155, 122368, 123152, 122265, 139815, 141664, 153176, 142310, 168730, 171856, 176336, 176272, 180848, 185368, 203432, 196724, 202444, 280540, 365084, 308620, 389924

Offset: 1

Omar E. Pol, Jul 09 2015

nonn

A pair of numbers x and y is called amicable if the sum of the proper divisors (or aliquot parts) of either one is equal to the other.
By definition a property of the amicable pair (x, y) is that x + y = sigma(x) = sigma(y).
The amicable pairs (x < y) are adjacent to each other in the list.
Also A260086 and A260087 interleaved.
Another version (A259180) lists the amicable pairs (x < y) ordered by increasing x.
Amicable numbers A063990 are the terms of this sequence in increasing order.
First differs from both A063990 and A259180 at a(17).

			-----------------------------------
       Amicable pair         Sum
          x      y          x + y
-----------------------------------
n     A260086 A260087      A259953
-----------------------------------
1        220     284          504
2       1184    1210         2394
3       2620    2924         5544
4       5020    5564        10584
5       6232    6368        12600
6      10744   10856        21600
7      12285   14595        26880
8      17296   18416        35712
9      66928   66992       133920
10     67095   71145       138240
11     63020   76084       139104
12     69615   87633       157248
...      ...     ...          ...
32    609928  686072      1296000
33    643336  652664      1296000
...
The sum of the proper divisors (or aliquot parts) of 220 is 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284. On the other hand the sum of the proper divisors (or aliquot parts) of 284 is 1 + 2 + 4 + 71 + 142 = 220. Note that 220 + 284 = sigma(220) = sigma(284) = 504. The sum 220 + 284 = 504 is the smallest sum of an amicable pair, so a(1) = 220 and a(2) = 284.
Note that some pairs (x, y) share the same sum (x + y), for example: (609928 + 686072) = (643336 + 652664) = sigma(609928) = sigma(686072) = sigma(643336) = sigma(652664) = 1296000, thus in the list first appears the pair (609928, 686072) and then (643336, 652664) because 609928 < 643336.

Laszlo Hars, Performance compared to mathematica Julia-users (2014)
Khelleos, Amicable numbers, CyberForum.ru (2011)
OEIS Wiki, Amicable numbers (This page needs work)
Wikipédia, Barátságos számok (contains a mistake: A063990 should be replaced with A259933)

Cf. A000203, A063990, A259180, A259953, A260086, A260087.

a(2n-1) + a(2n) = A000203(a(2n-1)) = A000203(a(2n)) = A259953(n).

A180164 The sum of the two numbers in an amicable pair, A002025(n) + A002046(n).

Original entry on oeis.org

504, 2394, 5544, 10584, 12600, 21600, 26880, 35712, 139104, 133920, 138240, 157248, 168480, 224640, 262080, 245520, 294840, 311040, 348192, 357120, 388800, 399168, 645624, 698544, 749952, 756000, 892800, 955206, 1017792, 1048320

Offset: 1

T. D. Noe, Aug 14 2010

nonn

This sequence initially shares many terms with A161005 because small amicable pairs are sometimes consecutive terms in the sorted list of amicable numbers, A063990.
This sequence is sorted by the smaller (abundant) member from A002025, so a(n) is not increasing. - Jeppe Stig Nielsen, Jan 27 2015
Duplicates occur, e.g., a(32)=a(35)=1296000. - Jeppe Stig Nielsen, Jan 27 2015
Comment originally by M. F. Hasler, Dec 14 2013, in A161005: "Also: The common value of sigma(a) = sigma(b) of the amicable pairs (a,b). See A137231 for the analog for amicable triples, and A116148 for quadruples." - Jeppe Stig Nielsen, Jan 27 2015
It is not known if a(n) is always even (see Hagis links). - Jeppe Stig Nielsen, Jan 31 2015
Are all terms abundant (A005101)? The first 10000 terms are. - Ivan N. Ianakiev, Apr 15 2021

			a(9) = A002025(9) + A002046(9) = 63020 + 76084 = 139104.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Peter Hagis, Lower bounds for relatively prime amicable numbers of opposite parity, Math. Comp. 24 (1970), 963-968.
Peter Hagis, Relatively Prime Amicable Numbers of Opposite Parity, Mathematics Magazine, Vol. 43, No. 1 (Jan., 1970), pp. 14-20.
Eric W. Weisstein's World of Mathematics, Pair Sum.

Cf. A002025, A002046, A066539, A259180 (amicable pairs).

s[n_] := DivisorSigma[1,n]-n; smallAmicableQ[n_] := Module[{b=s[n]}, n

Formula

a(n) = A259180(2n-1) + A259180(2n). - Omar E. Pol, Oct 22 2017

cp's OEIS Frontend

A328009 Irregular array read by rows in which row n lists the divisors of the n-th term of the sequence of amicable pairs (A259180).

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A001065 Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.

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A063990 Amicable numbers.

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A002025 Smaller of an amicable pair: (a,b) such that sigma(a) = sigma(b) = a+b, a < b.

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A002046 Larger of amicable pair.

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A259933 Amicable pairs (x < y) ordered by nondecreasing sum (x + y) and then by increasing x.

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