A125490 -id:A125490 - cp's OEIS frontend

A137231 Amicable triples. Sequence gives sigma values: A125490(n) + A125491(n) + A125492(n).

6552, 30240, 70680, 87360, 120960, 120960, 120960, 138240, 157248, 157248, 161280, 196560, 211680, 229320, 241920, 241920, 241920, 241920, 262080, 280800, 290160, 302400, 338688, 362880, 362880, 393120, 393120, 446400, 446400, 483840

Offset: 1

N. J. A. Sloane, Nov 27 2008, at the suggestion of R. K. Guy

nonn

Or, apply n -> sigma(n) to any of A125490-A125492.

John Cerkan, Table of n, a(n) for n = 1..10000 [Terms 1 through 1000 were computed by Donovan Johnson]

Cf. A125490-A125492.

a(13)-a(30) from Donovan Johnson, Apr 14 2010

Name edited by John Cerkan, Jun 18 2016

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

Original entry on oeis.org

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

A036471 Four numbers (a,b,c,d) with aA036472, A036473, A036474, A116148) respectively.

Original entry on oeis.org

3270960, 3767400, 4651920, 4969440, 5682600, 5405400, 6514200, 6126120, 6126120, 6320160, 6977880, 7013160, 6819120, 6966960, 7706160, 7731360, 7469280, 7469280, 8353800, 8288280, 8316000, 9258480, 9009000, 10048500, 9840600, 9923760, 9563400

Offset: 1

Yasutoshi Kohmoto

nonn

The sequence also contains the numbers 342151462276356306033089201934180, 6060968760324025992586151577119760, 99321681620793716265605322618607680, 409974875795109961540320351530842460160
The corresponding terms of A036472 would be: 357033595420351636473024008945820, 6324595118874800417522139587040240, 103641752270593503616169255168272320, 427807038915744951493564658435688099840
The corresponding terms of A036473 would be: 410581874773947356773179298713756, 7273164638852781748553461862929392, 119186052791523004137585762140907456, 491969994791994923787572902146102915072
The corresponding terms of A036474 would be: 428440439768072020710436590182244, 7589516361605847224013448168942608, 124370139086960335154801021607188544, 513368596793092084536932368118163836928
As stated, we first order by common sigma value. When the common value of sigma is the same for several quadruples, these are then sorted (ascending) by the smallest member. When the smallest members also agree, we go on to the second smallest members, and so on, lexicographically. - Jeppe Stig Nielsen, Feb 02 2015
Of the first 2000 quadruples, there are 78 cases where both the value of sigma, the value of the smallest member (a), and the value of the second smallest member (b) agree with those of the previous quadruple. The first time this happens is for n=17 and n=18 which correspond to the tuples (7469280, 8157240, 8873760, 9368520) and (7469280, 8157240, 9098460, 9143820), respectively. - Jeppe Stig Nielsen, Feb 02 2015

Donovan Johnson, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Amicable Quadruple

Cf. A036472, A036473, A036474, A116148.
Cf. A125490 - A125492 and A137231 (amicable triples).
Cf. A233553 for amicable 5-tuples.

The present first term was found by Dean Hickerson, Nov 06 2006
That this is the first term was confirmed by Giovanni Resta, Nov 14 2006, who also found a(2)-a(18).
Edited by N. J. A. Sloane, Nov 07 2006 and Nov 27 2006

A125492 Amicable triples: numbers such that sigma(x) = sigma(y) = sigma(z) = x+y+z, x

Original entry on oeis.org

2556, 11556, 27312, 32136, 42168, 41952, 41412, 54870, 55890, 57834, 54516, 70110, 75540, 83772, 88480, 84348, 84504, 83538, 102258, 100890, 100656, 108324, 138828, 132792, 132396, 143370, 146268, 168150, 162384, 189948

Offset: 1

Yasutoshi Kohmoto, Dec 11 2006

nonn

Giovanni Resta computed the first 12 terms.

As stated, we first order by common sigma value. When the common value of sigma is the same for several triples, these are then sorted (ascending) by the smallest member. When the smallest members also agree, we go on to the second smallest members, and so on, lexicographically. - John Cerkan, Jun 18 2016

			2556 is in the sequence since sigma(1980) = sigma(2016) = sigma(2556) = 6552 = 1980 + 2016 + 2556. - _Michael B. Porter_, Jun 29 2016

John Cerkan, Table of n, a(n) for n = 1..10000 [Terms 1 through 1000 were computed by Donovan Johnson]

Cf. A125490, A125491, A137231 (x+y+z).

Cf. A000203 (sigma function).

a(1)-a(12) from Giovanni Resta
Definition corrected by N. J. A. Sloane, Nov 27 2008
a(13)-a(30) from Donovan Johnson, Apr 14 2010

A036474 Amicable quadruples: the numbers d referred to in A036471.

Original entry on oeis.org

3834000, 4240800, 5089392, 5666760, 6939360, 7142100, 7932240, 8076960, 7976160, 8122464, 7738080, 7738920, 8136240, 8136240, 9121680, 9312480, 9368520, 9143820, 10159800, 10260180, 10403820, 10455984, 11179980, 10473120, 10930320, 10794960

Offset: 1

Yasutoshi Kohmoto

nonn

Donovan Johnson, Table of n, a(n) for n = 1..2000

Cf. A036471, A036472, A036473, A116148.
Cf. A002025, A002046 and A161005 for amicable pairs.
Cf. A125490 - A125492 and A137231 for amicable triples.
Cf. A233553 for amicable quintuples.

The present first term was found by Dean Hickerson, Nov 06 2006.
That this is the first term was confirmed by Giovanni Resta, Nov 14 2006, who also found a(2)-a(18).
Edited by N. J. A. Sloane, Nov 07 2006 and Nov 27 2006

A233553 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5. Groups of 5 subsequent terms list the five members in increasing size.

Original entry on oeis.org

53542288800, 59509850400, 59999219280, 60074174160, 61695597600, 67154527440, 68763895200, 69626138400, 71957405520, 72598125600, 67509842400, 72747675000, 73605331800, 75710489400, 78953074200, 87113426400, 88410722400, 89398663200, 96058282320, 96369633360

Offset: 1

M. F. Hasler, Dec 12 2013

The 5-tuple starting with 53542288800 was given by Donovan Johnson. The common value of sigma(x) is 294821130240.
A larger 5-tuple, (55766707476480, 56992185169920, 57515254917120, 57754372515840, 57829096765440), was found by Michel Marcus on Dec 09 2013. The common value of sigma(x) is 285857616844800.
A still larger example (227491164588441600, 228507506351308800, 229862628701798400, 230878970464665600, 243752632794316800), probably the first one to be published, had been found by Yasutoshi Kohmoto in 2008, cf. link to SeqFan post.
There are different definitions for amicable k-tuples, cf. link to MathWorld.

John Cerkan, Table of n, a(n) for n = 1..45
John Cerkan, More terms, with gaps.
Yasutoshi Kohmoto, Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v, SeqFan list, Nov 23 2008
Yasutoshi Kohmoto, Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v, SeqFan list, Dec 09 2013
Eric Weisstein's World of Mathematics, Amicable Triple.

Cf. A273928, A273930, A273931, A273933, A273934, A273936 (5-tuples).
Cf. A036471 - A036474 and A116148 (quadruples).
Cf. A125490 - A125492 and A137231 (triples).

More terms from John Cerkan, Jun 05 2016

A259303 Numbers n such that there exists an x!=n that makes {x,x,n} an amicable multiset.

Original entry on oeis.org

1560, 7380, 20448, 20664, 22752, 23352, 25248, 26964, 35328, 51360, 100200, 104412, 108792, 135540, 156312, 192096, 232740, 252252, 278460, 310752, 362160, 401688, 492660, 543456, 548592, 588744, 659736, 732060, 767520, 826848, 998712, 999648, 1129440, 1129464

Offset: 1

Jeppe Stig Nielsen, Jun 23 2015

nonn

We call the multiset {x,x,n} amicable iff sigma(x)=sigma(n)=x+x+n.
For the x values, see A259302.
If the condition x!=n were dropped, the terms from A005820 would also belong here.

John Cerkan, Table of n, a(n) for n = 1..1742

Cf. A005820, A259307, A259302, A259304, A259305, A259306, A125490.

isOK(n) = x=(sigma(n)-n)/2; x>0 && x!=n && denominator(x)==1 && sigma(x)==x+x+n;

A259302 Numbers n such that there exists an x!=n that makes {n,n,x} an amicable multiset.

Original entry on oeis.org

1740, 7776, 19260, 20640, 21384, 21924, 22428, 25830, 31440, 55968, 93024, 101010, 106140, 143910, 151164, 198792, 246510, 309582, 326196, 411138, 421596, 428256, 499464, 590112, 639288, 697158, 870552, 941094, 958716, 1060956, 1068210, 1087776, 1100640

Offset: 1

Jeppe Stig Nielsen, Jun 23 2015

nonn

We call the multiset {n,n,x} amicable iff sigma(n)=sigma(x)=n+n+x.
For the x values, see A259303.
If the condition x!=n were dropped, the terms from A005820 would also belong here.

John Cerkan, Table of n, a(n) for n = 1..1689

Cf. A005820, A259307, A259303, A259304, A259305, A259306, A125490.

isOK(n) = x=sigma(n)-2*n; x>0 && x!=n && sigma(x)==n+n+x;

A125491 Amicable triples: numbers such that sigma(x) = sigma(y) = sigma(z) = x+y+z, x

Original entry on oeis.org

2016, 9504, 22200, 31284, 41412, 40068, 40608, 43890, 54990, 52290, 53676, 69030, 69420, 77688, 83720, 84252, 82236, 80892, 87750, 93870, 96048, 97416, 107040, 129108, 129504, 137610, 127908, 156450, 144936, 152520

Offset: 1

Yasutoshi Kohmoto, Dec 11 2006

nonn

Giovanni Resta computed these terms.

As stated, we first order by common sigma value. When the common value of sigma is the same for several triples, these are then sorted (ascending) by the smallest member. When the smallest members also agree, we go on to the second smallest members, and so on, lexicographically. - John Cerkan, Jun 18 2016

			2016 is in the sequence since sigma(1980) = sigma(2016) = sigma(2556) = 6552 = 1980 + 2016 + 2556. - _Michael B. Porter_, Jun 29 2016

John Cerkan, Table of n, a(n) for n = 1..10000 [Terms 1 through 1000 were computed by Donovan Johnson]

Cf. A125490, A125492, A137231 (x+y+z).

Cf. A000203 (sigma function).

Definition corrected by N. J. A. Sloane, Nov 27 2008

a(13)-a(30) from Donovan Johnson, Apr 14 2010

A273928 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

Original entry on oeis.org

53542288800, 67154527440, 67509842400, 87113426400, 87502615200, 86133247200, 86133247200, 86133247200, 98471252880

Offset: 1

John Cerkan, Jun 04 2016

The 5-tuple starting with 53542288800 was given by Donovan Johnson. The common value of sigma(x) is 294821130240.
A larger 5-tuple, (55766707476480, 56992185169920, 57515254917120, 57754372515840, 57829096765440), was found by Michel Marcus on Dec 09 2013. The common value of sigma(x) is 285857616844800.
A still larger example (227491164588441600, 228507506351308800, 229862628701798400, 230878970464665600, 243752632794316800), probably the first one to be published, had been found by Yasutoshi Kohmoto in 2008, cf. link to SeqFan post.
Other terms from John Cerkan.
There are different definitions for amicable k-tuples, cf. link to MathWorld.

John Cerkan, More terms with gaps.
Yasutoshi Kohmoto, Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v, SeqFan list, Nov 23 2008
Yasutoshi Kohmoto, Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v, SeqFan list, Dec 09 2013
Eric W. Weisstein, Amicable Triple. From MathWorld--A Wolfram Web Resource.

Cf. A233553, A273930, A273931, A273933, A273934, A273936 (5-tuples).
Cf. A036471 - A036474 and A116148 (quadruples).
Cf. A125490 - A125492 and A137231 (triples).

cp's OEIS Frontend

A137231 Amicable triples. Sequence gives sigma values: A125490(n) + A125491(n) + A125492(n).

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

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A036471 Four numbers (a,b,c,d) with aA036472, A036473, A036474, A116148) respectively.

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A125492 Amicable triples: numbers such that sigma(x) = sigma(y) = sigma(z) = x+y+z, x

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A036474 Amicable quadruples: the numbers d referred to in A036471.

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A233553 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5. Groups of 5 subsequent terms list the five members in increasing size.

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A259303 Numbers n such that there exists an x!=n that makes {x,x,n} an amicable multiset.

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A259302 Numbers n such that there exists an x!=n that makes {n,n,x} an amicable multiset.

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A125491 Amicable triples: numbers such that sigma(x) = sigma(y) = sigma(z) = x+y+z, x

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