A259180 Amicable pairs.

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

Offset: 1

Author

Omar E. Pol, Jun 20 2015

Keywords

• nonn

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).

Examples

			  ------------------------------------
         Amicable pair          Sum
            x      y           x + y
  ------------------------------------
   n    A002025 A002046      A180164
  ------------------------------------
   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.
		

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

Crossrefs

Cf. A000203, A001065, A002025, A002046, A063990, A066539, A180164, A180202, A259933.

Programs

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 *)

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

Formula

a(2n-1) = A002025(n); a(2n) = A002046(n).

a(2n-1) + a(2n) = A000203(a(2n-1)) = A000203(a(2n)) = A180164(n).