A125492 -id:A125492 - cp's OEIS frontend

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

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).

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

A386933 Integers z such that there exist two integers 0

Original entry on oeis.org

81900, 161700, 163800, 175350, 245700, 261660, 323400, 327600, 350700, 409500, 485100, 490770, 491400, 499380, 523320, 526050, 526260, 573300, 646800, 647010, 655200, 671370, 701400, 702450, 737100, 784980, 808500, 819000, 876750, 970200, 971880, 981540, 982800, 990150, 998760

Offset: 1

S. I. Dimitrov, Aug 09 2025

The numbers x, y and z form a psi-amicable triple.

			163800 is in the sequence since psi(158340) = psi(161700) = psi(163800) = 564480 = 158340 + 161700 + 163800. Other examples: (322140, 322140, 323400), (14127960, 14224980, 14224980).

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

Cf. A001615, A125492, A323330, A385852, A386901.

A233626 Least member of an amicable n-tuple: (x[1],...,x[n]) such that sigma(x[1])=...=sigma(x[n])=x[1]+...+x[n], x[i]

Original entry on oeis.org

1, 220, 1980, 3270960, 53542288800

Offset: 1

M. F. Hasler, Dec 12 2013

Some authors use other definitions for amicable k-tuples, cf. link to MathWorld.

Eric W. Weisstein, Amicable Triple. From MathWorld--A Wolfram Web Resource.

Cf. A002025, A002046 and A161005 for amicable pairs.
Cf. A125490 - A125492 and A137231 for amicable triples.
Cf. A036471 - A036474 and A116148 for amicable quadruples.
Cf. A233553 for amicable quintuples.

A385155 Numbers z such that there exist two integers 0

Original entry on oeis.org

1380, 1540, 1560, 1638, 2016, 2250, 2520, 2556, 2700, 2772, 3024, 3120, 3312, 3360, 3408, 3480, 3640, 3654, 3780, 3816, 3828, 3876, 4200, 4320, 4440, 4452, 4620, 4920, 4956, 5220, 5280, 5292, 5304, 5340, 5400, 5460, 5472, 5640, 5700, 5724, 5760, 5940, 6048, 6060, 6180

Offset: 1

S. I. Dimitrov, Jun 19 2025

The numbers x, y and z form a HM(1,3)-amicable triple (HM = harmonic mean). See Dimitrov link. An amicable triple forms a HM(1,3)-amicable triple, so the larger member of an amicable triple A125492 is a term of this sequence.
From David A. Corneth, Jun 20 2025: (Start)
Let sx = sigma(x), sy = sigma(y) and sz = sigma(z).
We may write (1/sx + 1/sy + 1/sz)*(x + y + z) = (1/sx + 1/sy) * (x + y) + 1/sz * (x + y + z) + z * (1/sx + 1/sy). As z > y > x we have 2 * z > x + y so z > (x + y) / 2.
Therefore we have 3 = (1/sx + 1/sy) * (x + y) + 1/sz * (x + y + z) + z * (1/sx + 1/sy) > 1.5*(1/sx + 1/sy) * (x + y) + 1/sz * (x + y + z) > 1.5*(1/sx + 1/sy) * (x + y) and so (1/sx + 1/sy) * (x + y) < 2. Possibly 2 could be tightened due to 1/sz * (x + y + z) which is discarded for now.
If we know (1/sx + 1/sy) * (x + y) < U for some U then similarly x/s(x) can be tightened to 0.5*U and maybe more due to term 1/sy * (x + y).
Furthermore 3 = (1/sx + 1/sy + 1/sz)*(x + y + z) > 1/sx * (x + y + z) > 1/sx * (x + y + y) = 1/sx * (x + 2*y) which constraints y and later on z once 1/sx is known.
For every pair (k, m) in {(x, y), (x, z), (y, z)} from solutions (x, y, z) where z <= 10000 we have (k + m) * (1/sigma(k) + 1/sigma(m)) <= 1.5. Is that the case for every solution? (End).

			(840, 1020, 1380) is such a triple because (1/sigma(840) + 1/sigma(1020) + 1/sigma(1380))*(840 + 1020 + 1380) = 3.

David A. Corneth, PARI program
S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Cf. A000203, A125492, A384814, A384487.

PARI
```
\\ See Corneth link
```

Corrected and extended by David A. Corneth, Jun 20 2025

A233538 Triangle T(n,k) read by rows, which contains for 1<=k<=n the least amicable n-tuple T(n,1),..., T(n,n) such that sigma(T(n,k)) = T(n,1)+...+T(n,n).

Original entry on oeis.org

1, 220, 284, 1980, 2016, 2556, 3270960, 3361680, 3461040, 3834000, 53542288800, 59509850400, 59999219280, 60074174160, 61695597600

Offset: 1

Michel Marcus, M. F. Hasler, Dec 11 2013

Like amicable pairs, amicable n-tuples can be regular or irregular (see Pedersen link). The first amicable pair is regular. Then the first n-tuples are irregular.
For n=3 to 5, the first regular n-tuples are: [230880, 267168, 306336], [6966960, 7054320, 7840560, 8136240], [55766707476480, 56992185169920, 57515254917120, 57754372515840, 57829096765440].
On the other hand, for n>2, a n-tuple can be "very" irregular, that is, when the values of sigma(n-tuple[i]/GCD(n-tuple)) are all different. The first such n-tuples are [21168, 22200, 27312], [3767400, 4090320, 4150440, 4240800].
When n=2, irregular and "very irregular" is the same thing. The first irregular amicable pair is (1184, 1210) (see difference between A002025 and A215491).
Regular n-tuples can be found with the method described in the second Kohmoto link. Then it is eventually possible to derive another n-tuple using the same "seed". For this, it suffices to find an integer g' such that sigma(g')/g' = sigma(g)/g and coprime to the terms of the n-tuple divided by g.
The 6th row is smaller than (379952828833009557565440000, 387198605857900590673920000, 388674597474082097418240000, 388808778530098598031360000, 389307165309588457451520000, 393332596990083475845120000).

			Triangle begins:
1;
220, 284;                                 i.e. A002025(1), A002046(1).
1980, 2016, 2556;                         i.e. A125490(1), A125491(1), A125492(1).
3270960, 3361680, 3461040, 3834000;
53542288800, 59509850400, 59999219280, 60074174160, 61695597600.

Donovan Johnson, Re: A_k and RMPN, SeqFan list, Dec 12 2013
Yasutoshi Kohmoto, Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v, SeqFan list, Nov 23 2008
Yasutoshi Kohmoto, A_k and RMPN, SeqFan list, Dec 09 2013
J. M. Pedersen, Type system of amicable pairs
Eric Weisstein's World of Mathematics, Amicable Pair
Eric Weisstein's World of Mathematics, Amicable Triple
Eric Weisstein's World of Mathematics, Amicable Quadruple

Cf. A233626 (first column).
Cf. A002025, A002046, A161005, (amicable pairs).
Cf. A125490 - A125492, A137231, (amicable triples).
Cf. A036471 - A036474, A116148, (amicable quadruples).
Cf. A233553, A233626 (first row).

A385749 Numbers z such that there exist two integers 0

Original entry on oeis.org

120, 672, 1740, 2556, 4680, 11556, 27312, 32136, 41412, 41952, 42168

Offset: 1

S. I. Dimitrov, Jul 08 2025

The numbers x, y and z form a WHM(2)-amicable triple (WHM = weighted harmonic mean). An amicable triple forms a WHM(2)-amicable triple, so the larger member of an amicable triple A125492 is a term of this sequence.

			(1980, 2016, 2556) is such a triple because (1980^2/sigma(1980)^2 + 2016^2/sigma(2016)^2 + 2556^2/sigma(2556)^2)*(1980 + 2016 + 2556)^2 = 1980^2 + 2016^2 + 2556^2.
Other kinds of triples are: (120,120,120), (1560, 1740, 1740) and (117, 117, 4680).
Note that (117, 117, 4680) is the only known WHM(2)-amicable triple that is not an amicable triple.

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

Cf. A000203, A125492, A384487.

Cf. A005820 (a subsequence, for (x,x,x) triples).

A386010 Numbers z such that there exist two integers 0

Original entry on oeis.org

120, 672, 1188, 1740, 2556, 11172, 11556, 11628, 27312, 32136, 41412, 41952, 42168

Offset: 1

S. I. Dimitrov, Jul 14 2025

The numbers x, y and z form a GM-amicable triple (GM = Geometric Mean). See Dimitrov link. An amicable triple forms a GM-amicable triple, so the larger member of an amicable triple A125492 is a term of this sequence.

			(1080, 1092, 1188) is such a triple because sigma(1080)*sigma(1092)*sigma(1188) = (1080 + 1092 + 1188)^3.

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

Cf. A000203, A125492, A383932.

cp's OEIS Frontend

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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A384487 Numbers k such that there exist two integers 0

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A386933 Integers z such that there exist two integers 0

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A233626 Least member of an amicable n-tuple: (x[1],...,x[n]) such that sigma(x[1])=...=sigma(x[n])=x[1]+...+x[n], x[i]

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A385155 Numbers z such that there exist two integers 0

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A233538 Triangle T(n,k) read by rows, which contains for 1<=k<=n the least amicable n-tuple T(n,1),..., T(n,n) such that sigma(T(n,k)) = T(n,1)+...+T(n,n).

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A385749 Numbers z such that there exist two integers 0

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A386010 Numbers z such that there exist two integers 0

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