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).

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

Offset

1,2

Comments

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).

Links

Table of n, a(n) for n=1..15.

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

Example

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.

Crossrefs

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).

Sequence in context: A262624 A262622 A287026 * A328063 A328065 A287011

Adjacent sequences: A233535 A233536 A233537 * A233539 A233540 A233541

Keyword

nonn,tabl,hard,more

Author

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

Status

approved