A125491 -id:A125491 - 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

A125490 Amicable triples: numbers such that sigma(x) = sigma(y) = sigma(z) = x+y+z, x

Original entry on oeis.org

1980, 9180, 21168, 23940, 37380, 38940, 38940, 39480, 46368, 47124, 53088, 57420, 66720, 67860, 69720, 73320, 75180, 77490, 72072, 86040, 93456, 96660, 92820, 100980, 100980, 112140, 118944, 121800, 139080, 141372

Offset: 1

Yasutoshi Kohmoto, Dec 11 2006

nonn

G. 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

			1980 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. A125491, A125492, A137231 (x+y+z).
Cf. A036471 - A036474 and A116148 (amicable quadruples).
Cf. A233553 for amicable 5-tuples.
Cf. A000203 (sigma function).

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

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

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

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

A386901 Integers y such that there exist two integers 0

Original entry on oeis.org

80850, 158340, 161070, 161700, 232050, 242550, 316680, 322140, 323400, 404250, 464100, 474810, 475020, 483210, 485100, 485940, 565950, 633360, 641550, 644280, 646800, 662340, 696150, 727650, 791700, 805350, 808500, 963270, 966420, 967890, 970200, 971880

Offset: 1

S. I. Dimitrov, Aug 07 2025

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

			158340 is in the sequence since psi(150150) = psi(158340) = psi(175350) = 483840 = 150150 + 158340 + 175350. Other examples: (232050, 232050, 261660), (7091700, 7098630, 7098630).

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

Cf. A001615, A125491, A323329, A385852, A386933.

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).

cp's OEIS Frontend

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

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A125490 Amicable triples: numbers such that sigma(x) = sigma(y) = sigma(z) = x+y+z, x

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

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A384487 Numbers k such that there exist two integers 0

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A386901 Integers y 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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