cp's OEIS Frontend

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

%I A002025 M5414 N2352 #115 Jul 02 2025 11:34:49
%S A002025 220,1184,2620,5020,6232,10744,12285,17296,63020,66928,67095,69615,
%T A002025 79750,100485,122265,122368,141664,142310,171856,176272,185368,196724,
%U A002025 280540,308620,319550,356408,437456,469028,503056,522405,600392,609928
%N A002025 Smaller of an amicable pair: (a,b) such that sigma(a) = sigma(b) = a+b, a < b.
%C A002025 Sometimes called friendly numbers, but this usage is deprecated.
%C A002025 All terms are abundant (A005101). - _Michel Marcus_, Mar 10 2013
%C A002025 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
%C A002025 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
%C A002025 For the related amicable pairs see A259180. - _Omar E. Pol_, Jul 15 2015
%C A002025 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
%D A002025 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.
%D A002025 Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 48-49.
%D A002025 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002025 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002025 T. D. Noe and Sergei Chernykh, <a href="/A002025/b002025.txt">Table of n, a(n) for n = 1..415523</a> [All terms up to 10^17. Terms 39375 through 415523 were computed by Sergei Chernykh]
%H A002025 J. Alanen, O. Ore and J. Stemple, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0222006-7">Systematic computations on amicable numbers</a>, Math. Comp., 21 (1967), 242-245.
%H A002025 J. Bell, <a href="http://arXiv.org/abs/math.NT/0409196">A translation of Leonhard Euler's...</a>, arXiv:math/0409196 [math.HO], 2004-2009.
%H A002025 W. Borho and H. Hoffmann, <a href="http://dx.doi.org/10.1090/S0025-5718-1986-0815849-1">Breeding Amicable Numbers in Abundance</a>, Math. Comp., 46 (1986), 281-293.
%H A002025 S. Chernykh, <a href="http://sech.me/ap/">Amicable pairs list</a>.
%H A002025 Paul Erdős, <a href="http://www.renyi.hu/~p_erdos/1955-03.pdf">On amicable numbers</a>, Publ. Math. Debrecen 4 (1955), pp. 108-111.
%H A002025 E. B. Escott, <a href="/A002025/a002025.pdf">Amicable numbers</a>, Scripta Mathematica, 12 (1946), 61-72. [Annotated scanned copy]
%H A002025 L. Euler, De numeris amicabilibus, Opuscula varii argumetii, pages 23-107, 1750. Reprinted in <a href="http://gallica.bnf.fr/scripts/ConsultationTout.exe?E=0&amp;O=N006952">Opera mathematica: Series prima. Volumen II, Leonhardi Euleri commentationes arithmeticae</a>. Sub ausp. soc. scient. nat. Helv., Teubner, Leipzig, Series I, Vol. 1915, pp. 86-162.
%H A002025 M. Garcia, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/GARCIA/millionc.html">A Million New Amicable Pairs</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.6.
%H A002025 Mariano García, Jan Munch Pedersen, and Herman J. J. te Riele, <a href="https://web.archive.org/web/20140531174057/http://oai.cwi.nl/oai/asset/4143/04143D.pdf">Amicable pairs, a survey</a>, Report MAS-R0307, 2003, Centrum Wiskunde en Informatica.
%H A002025 Mariano García, Jan Munch Pedersen, and Herman J. J. te Riele, <a href="https://core.ac.uk/download/pdf/301650336.pdf">Amicable pairs, a survey</a>, Fields Institute Comm. (2004) Vol. 41.
%H A002025 S. S. Gupta, <a href="http://www.shyamsundergupta.com/amicable.htm">Amicable Numbers</a>.
%H A002025 E. J. Lee, <a href="http://dx.doi.org/10.1090/S0025-5718-1968-0224543-9">Amicable Numbers and the Bilinear Diophantine Equation</a>, Math. Comp., 22 (1968), 181-187.
%H A002025 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math09/ami02.htm">First 236 amicable pairs</a>.
%H A002025 D. Moews, <a href="http://djm.cc/amicable.html">Perfect, amicable and sociable numbers</a>.
%H A002025 Passawan Noppakaew and Prapanpong Pongsriiam, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Pongsriiam/pong43.html">Product of Some Polynomials and Arithmetic Functions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
%H A002025 J. O. M. Pedersen, <a href="http://amicable.homepage.dk/knwnap.htm">Known Amicable Pairs</a>. [Broken link]
%H A002025 J. O. M. Pedersen, <a href="http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a>. [Broken link]
%H A002025 J. O. M. Pedersen, <a href="http://web.archive.org/web/20140502102524/http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a>. [Via Internet Archive Wayback-Machine]
%H A002025 J. O. M. Pedersen, <a href="/A063990/a063990.pdf">Tables of Aliquot Cycles</a>. [Cached copy, pdf file only]
%H A002025 Carl Pomerance, <a href="https://www.math.dartmouth.edu/~carlp/Amicable1.pdf">On the distribution of amicable numbers</a>, J. reine angew. Math. 293/294 (1977), pp. 217-222.
%H A002025 Carl Pomerance, <a href="https://www.math.dartmouth.edu/~carlp/Amicable2.pdf">On the distribution of amicable numbers, II</a>, J. reine angew. Math. 325 (1981), pp. 183-188.
%H A002025 H. J. J. te Riele, <a href="http://dx.doi.org/10.1090/S0025-5718-1974-0330033-8">Four large amicable pairs</a>, Math. Comp., 28 (1974), 309-312.
%H A002025 H. J. J. te Riele, <a href="http://dx.doi.org/10.1090/S0025-5718-1986-0842142-3">Computation of all the amicable pairs below 10^10</a>, Math. Comp., 47 (1986), 361-368 and Supplement pp. S9-S40.
%H A002025 H. J. J. te Riele et al., <a href="http://oai.cwi.nl/oai/asset/6222/6222A.pdf">Table of Amicable Pairs between 10^10 and 10^52</a>, Note NM-N8603, Department of Numerical Mathematics, Centre for Mathematics and Computer Science, Amsterdam, 1986. (Warning: file size is 65MB.)
%H A002025 T. Trotter, Jr., <a href="https://web.archive.org/web/20101130191602/http://www.trottermath.net/numthry/amicable.html">Amicable Numbers</a>, archived from the original.
%H A002025 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AmicablePair.html">Amicable Pair</a>.
%F A002025 a(n) = A259180(2n-1) = A180164(n) - A259180(2n) = A180164(n) - A002046(n). - _Omar E. Pol_, Jul 15 2015
%t A002025 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_ *)
%o A002025 (PARI) aliquot(n)=sigma(n)-n
%o A002025 isA002025(n)={if (n>1, local(a);a=aliquot(n);a>n && aliquot(a)==n)} \\ _Michael B. Porter_, Apr 11 2010
%o A002025 (PARI) for(n=1,1e6,(s=sigma(n))>2*n && sigma(s-n)==s && print1(n",")) \\ _M. F. Hasler_, Dec 14 2013
%o A002025 (PARI) 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
%Y A002025 Cf. A000203, A002046, A063990, A066873, A180164, A259180.
%K A002025 nonn,nice
%O A002025 1,1
%A A002025 _N. J. A. Sloane_
%E A002025 More terms from Larry Reeves (larryr(AT)acm.org), Oct 24 2000