A036474 -id:A036474 - cp's OEIS frontend

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

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).

A036472 Amicable quadruples: the numbers b referred to in A036471.

Original entry on oeis.org

3361680, 4090320, 4839120, 5116320, 5831280, 6029100, 6640200, 6541920, 6922080, 6902280, 7087080, 7436520, 7327320, 7054320, 7722000, 7852320, 8157240, 8157240, 8920800, 9360540, 8759520, 9621360, 10174500, 10174500, 10098000, 10316880, 10085040

Offset: 1

Yasutoshi Kohmoto

nonn

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

Cf. A036471, A036473, A036474, A116148.

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

A036473 Amicable quadruples: the numbers c referred to in A036471.

Original entry on oeis.org

3461040, 4150440, 4918320, 5475960, 5920200, 6421800, 6768720, 7559640, 7280280, 7685496, 7227360, 7688520, 7715400, 7840560, 7948080, 8125920, 8873760, 9098460, 10063200, 9830520, 10260180, 9661680, 10521000, 10188360, 10914120, 10747440, 11241720

Offset: 1

Yasutoshi Kohmoto

nonn

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

Cf. A036471, A036472, A036474, A116148.

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

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

nonn

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.

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

isok(k) = my(s=sigma(k)); for (m=1, k-1, if ((sigma(m)==s) && (s==m+2*k), return(m))); \\ Michel 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

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.

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

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

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A036472 Amicable quadruples: the numbers b referred to in A036471.

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A036473 Amicable quadruples: the numbers c referred to in A036471.

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A383239 Integers k such that there exists an integer 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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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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