A002046 Larger of an amicable pair with smaller member A002025(n). (Formerly M5435 N2363)

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

1,1

Comments

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

References

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.

Links

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

S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

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]

Irving H. Siegel, Index-number differences: geometric means, Journal of the American Statistical Association, 37 (218): 271-274 (June 1942), doi:10.1080/01621459.1942.10500636

T. Trotter, Jr., Amicable Numbers

Eric Weisstein's World of Mathematics, Amicable Pair

Formula

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

Example

The smallest amicable pair (a, b), i.e., such that sigma(a) = sigma(b) = a + b, is (220 = 2^2*5*11, 284 = 2^2*71), with sigma(220) = sigma(284) = 504 = 220 + 284, therefore A002025(1) = 220 and a(1) = 284. - M. F. Hasler, Jun 08 2026

Maple

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

Mathematica

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 *)
(* Alternative: *)
Table[DivisorSigma[1, A002025[n]] - A002025[n], {n, 50}] (* T. D. Noe, Sep 20 2011 *)

Prog

(PARI) aliquot(n)=sigma(n)-n
isA002046(n)={if (n>1, local(a); a=aliquot(n); a<n && aliquot(a)==n)} \\ Michael B. Porter, Apr 17 2010

Crossrefs

Cf. A002025, A063990, A180164, A259180.

Sequence in context: A090789 A234970 A260087 * A339679 A339683 A343917

Adjacent sequences: A002043 A002044 A002045 * A002047 A002048 A002049

Keyword

nonn,nice,changed

Author

N. J. A. Sloane

Extensions

More terms from Larry Reeves, Oct 25 2000

Status

approved