cp's OEIS Frontend

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).

%I A233538 #44 Feb 16 2025 08:33:21
%S A233538 1,220,284,1980,2016,2556,3270960,3361680,3461040,3834000,53542288800,
%T A233538 59509850400,59999219280,60074174160,61695597600
%N 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).
%C A233538 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.
%C A233538 For n=3 to 5, the first regular n-tuples are: [230880, 267168, 306336], [6966960, 7054320, 7840560, 8136240], [55766707476480, 56992185169920, 57515254917120, 57754372515840, 57829096765440].
%C A233538 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].
%C A233538 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).
%C A233538 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.
%C A233538 The 6th row is smaller than (379952828833009557565440000, 387198605857900590673920000, 388674597474082097418240000, 388808778530098598031360000, 389307165309588457451520000, 393332596990083475845120000).
%H A233538 Donovan Johnson, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2013-December/012110.html">Re: A_k and RMPN</a>, SeqFan list, Dec 12 2013
%H A233538 Yasutoshi Kohmoto, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2008-November/000217.html">Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)=Sigma(v)=x+y+z+u+v</a>, SeqFan list, Nov 23 2008
%H A233538 Yasutoshi Kohmoto, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2013-December/012089.html">A_k and RMPN</a>, SeqFan list, Dec 09 2013
%H A233538 J. M. Pedersen, <a href="http://amicable.homepage.dk/apstat.htm#typesys">Type system of amicable pairs</a>
%H A233538 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AmicablePair.html">Amicable Pair</a>
%H A233538 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AmicableTriple.html">Amicable Triple</a>
%H A233538 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AmicableQuadruple.html">Amicable Quadruple</a>
%e A233538 Triangle begins:
%e A233538 1;
%e A233538 220, 284;                                 i.e. A002025(1), A002046(1).
%e A233538 1980, 2016, 2556;                         i.e. A125490(1), A125491(1), A125492(1).
%e A233538 3270960, 3361680, 3461040, 3834000;
%e A233538 53542288800, 59509850400, 59999219280, 60074174160, 61695597600.
%Y A233538 Cf. A233626 (first column).
%Y A233538 Cf. A002025, A002046, A161005, (amicable pairs).
%Y A233538 Cf. A125490 - A125492, A137231, (amicable triples).
%Y A233538 Cf. A036471 - A036474, A116148, (amicable quadruples).
%Y A233538 Cf. A233553, A233626 (first row).
%K A233538 nonn,tabl,hard,more
%O A233538 1,2
%A A233538 _Michel Marcus_, _M. F. Hasler_, Dec 11 2013