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A002025 Smaller of an amicable pair: (a,b) such that sigma(a)=sigma(b)=a+b, a<b.
(Formerly M5414 N2352)
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 (list; graph; refs; listen; history; text; internal format)



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)) members 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.

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.

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

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

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

Eric Weisstein's World of Mathematics, Amicable Pair.


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


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 *)


(PARI) aliquot(n)=sigma(n)-n

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

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

(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


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

Sequence in context: A257354 A102073 A234969 * A260086 A241615 A180219

Adjacent sequences:  A002022 A002023 A002024 * A002026 A002027 A002028




N. J. A. Sloane


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



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Last modified August 24 05:48 EDT 2017. Contains 291052 sequences.