A125490 -id:A125490 - cp's OEIS frontend

A273930 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

59509850400, 68763895200, 72747675000, 88410722400, 88021533600, 89894684880, 89894684880, 90391981680, 102481394400

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

A273931 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

Original entry on oeis.org

59999219280, 69626138400, 73605331800, 89398663200, 89398663200, 90391981680, 94320626400, 94832992800, 103169959200

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 Weisstein's World of Mathematics, Amicable Triple.

Cf. A233553, A273928, A273930, A273933, A273934, A273936 (5-tuples).
Cf. A036471 - A036474 and A116148 (quadruples).
Cf. A125490 - A125492 and A137231 (triples).

A273933 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

Original entry on oeis.org

60074174160, 71957405520, 75710489400, 96058282320, 96058282320, 97306569360, 96759542880, 94972878000, 109117562400

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

A273934 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

Original entry on oeis.org

61695597600, 72598125600, 78953074200, 96369633360, 96369633360, 103073639760, 99692021520, 100469023200, 109446377040

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

A273936 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

Original entry on oeis.org

294821130240, 350100092160, 368526412800, 457350727680, 457350727680, 466800122880, 466800122880, 466800122880, 522686545920

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

A383239 Integers k such that there exists an integer 0

Original entry on oeis.org

1740, 7776, 22428, 55968, 106140, 143910, 198792, 246510, 309582, 326196, 411138, 421596, 428256, 590112, 639288, 697158, 870552, 941094, 958716, 1060956, 1087776, 1105884, 1269828, 1341660, 1361568, 1447620, 1495494, 1512810, 1626324, 1727940, 1819392

Offset: 1

S. I. Dimitrov, Apr 20 2025

S. I. Dimitrov introduced the notion of (alpha_1,...,alpha_k)-multiamicable k-tuples.

The asymptotic density of (alpha_1, alpha_2)-multiamicable pairs relative to the positive integers is 0.

			For k=2, alpha_1=1, alpha_2=2 we have (1560, 1740), (7380, 7776), (20664, 22428), (543456, 590112), (588744, 639288),

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

Subsequence of A259302.

Cf. A000203, A005820, A063990, A259180, A125490, A036474, A259303, A292365.

isok(k) = my(s=sigma(k), m=s-2*k); m>0 && mMichel Marcus, Apr 28 2025

We say that the natural numbers n_1,..., n_k form an (alpha_1,...,alpha_k)-multiamicable k-tuple if sigma(n_1)=sigma(n_2)=...=sigma(n_k)=alpha_1n_1+alpha_2n_2+...+alpha_kn_k, where alpha_1,...,alpha_k are positive integers, where sigma(n) is the sum of the divisors of n.

More terms from Sean A. Irvine, May 04 2025

A384487 Numbers k such that there exist two integers 0

Original entry on oeis.org

396, 504, 600, 756, 840, 924, 1056, 1080, 1140, 1170, 1260, 1320, 1428, 1440, 1488, 1512, 1540, 1560, 1596, 1638, 1650, 1656, 1680, 1704, 1710, 1740, 1800, 1820, 1840, 1848, 1872, 1932, 1980, 2016, 2040, 2100, 2160, 2184, 2232, 2244, 2256, 2280, 2340, 2352, 2380, 2400, 2430, 2436, 2448, 2460, 2484

Offset: 1

S. I. Dimitrov, Jun 01 2025

nonn

The numbers i, j and k form a WHM(1)-amicable triple (WHM = weighted harmonic mean). See Dimitrov link.

			504 is a term because (72, 360, 504) is a triple with 72/sigma(72) + 360/sigma(360) + 504/sigma(504) = 1.
420 is not a term because the corresponding triple is (84, 420, 420).

Robert Israel, Table of n, a(n) for n = 1..600
S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Cf. A000203, A125490, A125491, A125492, A253534, A253535, A383964.

S:= {}: S2:= {}: R:= NULL: count:= 0:
for k from 1 while count < 100 do
  v:= k/numtheory:-sigma(k);
  if member(1-v,S2) then
    R:= R, k; count:= count+1;
 fi;
  S2:= S2 union map(t -> `if`(t+v<1,t+v,NULL),S);
  S:= S union {v};
od:
R; # Robert Israel, Jul 01 2025

isok(k) = for (i=1, k-1, for (j=i+1, k-1, if (i/sigma(i) + j/sigma(j) + k/sigma(k) == 1, /* print([i,j,k]); */ return(1)););); \\ Michel Marcus, Jun 02 2025

More terms from Michel Marcus, Jun 02 2025

A385852 Integers x such that there exist two integers 0

Original entry on oeis.org

79170, 150150, 158340, 161070, 232050, 237510, 300300, 316680, 322140, 395850, 450450, 464100, 468930, 474810, 475020, 483210, 554190, 570570, 600600, 622440, 633360, 641550, 644280, 696150, 712530, 750750, 791700, 805350, 937860, 949620, 950040, 963270, 966420

Offset: 1

S. I. Dimitrov, Aug 07 2025

The numbers x, y and z form a psi-amicable triple according to Dimitrov's definition.

			79170 is in the sequence since psi(79170) = psi(80850) = psi(81900) = 241920 = 79170 + 80850 + 81900. Other examples: (161070, 161070, 161700), (7063980, 7112490, 7112490).

S. I. Dimitrov, On psi-amicable numbers and their generalizations, arXiv:2508.02318 [math.NT], 2025.

Cf. A001615, A125490, A323329, A386901, A386933.

A255215 Numbers that belong to at least one amicable tuple.

Original entry on oeis.org

1, 220, 284, 1184, 1210, 1980, 2016, 2556, 2620, 2924, 5020, 5564, 6232, 6368, 9180, 9504, 10744, 10856, 11556, 12285, 14595, 17296, 18416, 21168, 22200, 23940, 27312, 31284, 32136, 37380, 38940, 39480, 40068, 40608, 41412, 41952, 42168, 43890, 46368, 47124

Offset: 1

Jeppe Stig Nielsen, Feb 17 2015

nonn

Call a finite set {x_1, x_2, ..., x_k} of natural numbers (the x_i are pairwise distinct) an amicable k-tuple iff sigma(x_1)=sigma(x_2)=...=sigma(x_k)=x_1+x_2+...+x_k. Here sigma=A000203. For k=1, the only possible amicable one-tuple is {1}. For k=2 we get the classical amicable pairs (A063990). k=3 is amicable triples (A125490), k=4 amicable quadruples (A036471), and so on. A natural number n belongs to this sequence if and only if n is a member of some amicable k-tuple.
By definition, this sequence contains no duplicates.
For k<>2, an amicable k-tuple is not an aliquot cycle.

			1 belongs to this sequence because {1} is considered an amicable one-tuple.
284 belongs to this sequence because {220, 284} is an amicable pair.
2016 belongs to this sequence because {1980, 2016, 2556} is an amicable triple.
38940 is included in this sequence only once even if both {38940, 40068, 41952} and {38940, 40608, 41412} are amicable.
1000 does not belong to this sequence. To prove that, note that sigma(1000)=2340. Then find all x such that sigma(x)=2340, these are 792, 1000, 1062, 1305, 1611, 1945, 2339. Run through all subsets of 792, 1000, 1062, 1305, 1611, 1945, 2339 that include 1000 to verify that no such subset has a sum of 2340.
A tuple (or multiset) like {1560, 1740, 1740} where some element(s) are repeated, is not allowed here, and neither 1560 nor 1740 belongs to this sequence.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..361

Cf. A000203, A063990, A125490, A036471.

Cf. A259307 (duplicates allowed in tuple).

(notSubsetSum(desiredSum, searchSet) = { /* strongly inspired by is_A006037 function from A006037 */ local(t); /* return nonzero iff desiredSum is not the sum of a subset of searchSet */ setsearch( Set(searchSet), desiredSum ) & return /* equal to one element of searchSet */; while( #searchSet & searchSet[ #searchSet]>desiredSum, searchSet=vecextract(searchSet, "^-1")); desiredSum >= (t = sum(i=1, #searchSet, searchSet[i])) & return( desiredSum-t /* nonzero if desiredSum>t */ ); desiredSum > searchSet[ #searchSet] & ! notSubsetSum( desiredSum - searchSet[ #searchSet], searchSet=vecextract( searchSet, "^-1" )) & return; notSubsetSum( desiredSum, searchSet ) }); (othersWithSameSigma(n) = { s=sigma(n); [ x | x<-[1..s-1] , sigma(x)==s&&x!=n ] }); (is_A255215(x) = !notSubsetSum(sigma(x)-x, othersWithSameSigma(x)))

A386726 Numbers x such that there exist two integers 0

Original entry on oeis.org

2, 238, 280, 308, 310, 382, 790, 795, 920, 952, 1034, 1162, 1246, 1330, 1410, 1434, 2002, 2024, 2506, 2632, 2728, 2750, 2926, 3040, 3210, 3452, 3496, 3500, 3630, 4134, 4260, 4466, 4550, 4968, 5080, 5278, 5396, 5520, 5530, 5756, 6128, 6230, 6426, 6888, 7288, 7584, 7640, 7910, 7990

Offset: 1

S. I. Dimitrov, Jul 31 2025

nonn

The numbers x, y and z form an amicable triple according to Yanney's definition.

			238 is in the sequence since sigma(238) = sigma(255) = sigma(371) = 432 = (238 + 255 + 371)/2.

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Benjamin Franklin Yanney, Another definition of amicable numbers and some of their relations to Dickson's amicables, Amer. Math. Monthly, Vol. 30, No. 6 (1923), 311-315.

Cf. A000203, A125490, A386727.

isok(x1) = my(s=sigma(x1), vx=select(x->(x>=x1), invsigma(s)), v=vector(3, i, vx[1])); for (i=1, #vx, v[2] = vx[i]; for (j=1, #vx, v[3] = vx[j]; if (vecsum(v) == 2*s, return(1)););); \\ Michel Marcus, Aug 01 2025

More terms from Michel Marcus, Aug 01 2025

cp's OEIS Frontend

A273930 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

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A273931 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

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A273933 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

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A273934 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

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A273936 Amicable 5-tuples: (x1,...,x5) such that sigma(x1)=...=sigma(x5)=x1+...+x5, x1

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A383239 Integers k such that there exists an integer 0

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A384487 Numbers k such that there exist two integers 0

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Maple

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A385852 Integers x such that there exist two integers 0

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A255215 Numbers that belong to at least one amicable tuple.

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PARI

A386726 Numbers x such that there exist two integers 0

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