OFFSET
1,1
COMMENTS
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.
This is A002025 and A002046 interleaved hence the amicable pairs (x < y), ordered by increasing x, are adjacent to each other in the list.
By definition a property of the amicable pair (x, y) is that x + y = sigma(x) = sigma(y).
Amicable numbers A063990 are the terms of this sequence in increasing order.
First differs from A063990 at a(18).
For another version see A259933.
First differs from A259933 at a(17).
LINKS
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]
S. Chernykh, Amicable Numbers
S. Chernykh, Amicable pairs list
G. D'Abramo, On Amicable Numbers With Different Parity, arXiv:math/0501402 [math.HO], 2005-2007.
E. B. Escott, Amicable numbers, Scripta Mathematica, 12 (1946), 61-72 [Annotated scanned copy]
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.
S. S. Gupta, Amicable Numbers
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
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
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
Juan Luis Varona, On the Solution of the Equation n = a*k + b*p_k by Means of an Iterative Method, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.5.
Gérard Villemin's Almanach of Numbers, Nombres amiables et sociables
Eric Weisstein's World of Mathematics, Amicable Pair
Wikipedia, Amicable number
FORMULA
EXAMPLE
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Amicable pair Sum
x y x + y
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------------------------------------
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 63020 76084 139104
10 66928 66992 133920
11 67095 71145 138240
12 69615 87633 157248
... ... ... ...
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 smallest amicable pair is (220, 284), so a(1) = 220 and a(2) = 284.
MATHEMATICA
f[n_] := Block[{s = {}, g, k}, g[x_] := DivisorSigma[1, x] - x; Do[k = g@ i; If[And[g@ k == i, k != i, ! MemberQ[s, i]], s = s~Join~{i, k}], {i, n}]; s]; f@ 300000 (* Michael De Vlieger, Jul 02 2015 *)
PROG
(PARI) A259180_upto(N, L=List(), s)={ forfactored(n=1, N, (s=sigma(n[2]))>2*n[1] && sigma(s-n[1])==s && listput(L, [n[1], s-n[1]])); concat(L)} \\ M. F. Hasler, Oct 11 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jun 20 2015
STATUS
approved