A066873 -id:A066873 - cp's OEIS frontend

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

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

Offset: 1

N. J. A. Sloane

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)) 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

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.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 48-49.
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.
Mariano García, Jan Munch Pedersen, and Herman J. J. te Riele, Amicable pairs, a survey, Report MAS-R0307, 2003, Centrum Wiskunde en Informatica.
Mariano García, Jan Munch Pedersen, and Herman J. J. te Riele, Amicable pairs, a survey, Fields Institute Comm. (2004) Vol. 41.
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.
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]
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, archived from the original.
Eric Weisstein's World of Mathematics, Amicable Pair.

Cf. A000203, A002046, A063990, A066873, A180164, A259180.

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 *)

aliquot(n)=sigma(n)-n
isA002025(n)={if (n>1, local(a);a=aliquot(n);a>n && aliquot(a)==n)} \\ Michael B. Porter, Apr 11 2010

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

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

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

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

The list starts with n=1.

Comparing with the numbers of even amicable pairs in A066873, up to 10^4, the proportion of odd amicable pairs is 0%; up to 10^5 it is 23% and up to 10^10 is 28.9%. Up to 10^15, it is 40.4% and up to 10^19 this percentage is 45.9%. It is possible that this trend holds true for more amicable pairs, and thus most amicable number pairs are odd.

cp's OEIS Frontend

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

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A290985 Number of amicable pairs with lesser member at most 2^n.

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A360054 Number of odd amicable pairs where the smaller term of the pair is less than 10^n.

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