%I #4 Jun 19 2020 04:12:54
%S 84,228,280,340,372,408,468,480,516,624,740,792,804,840,868,880,948,
%T 984,1012,1188,1200,1204,1236,1240,1364,1380,1440,1456,1488,1496,1524,
%U 1624,1652,1668,1672,1700
%N First elements of maximal isospectral chains of length 2.
%C Isospectral Chain Conjecture: There exist isospectral chains of any positive length.
%C A number N is the first element of a maximal isospectral chain of length n if it is not part of an isospectral chain of length greater than n.
%C Two integers are isospectral if they have the same spectral basis. An isospectral chain of length n is a sequence N1,...,Nn of integers with the same spectral basis such that N1=2*N2=...=n*Nn and index(Nk)=k. A chain is maximal if it cannot be extended to an isospectral chain of length n+1.
%C The spectral sum of an integer N with at least two prime factors is the sum of the elements of its spectral basis, and is of the form k*N+1, where k is a positive integer. Then we say that N has index k, index(N)=k.
%H Garret Sobczyk, <a href="https://garretstar.com/secciones/publications/docs/monthly336-346.pdf">The Missing Spectral Basis in Algebra and Number Theory</a>, The American Mathematical Monthly, Vol. 108, No. 4 (April 2001), pp. 336-346.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Idempotent_(ring_theory)">Idempotent (ring theory)</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Peirce_decomposition">Peirce decomposition</a>
%e a(1) = 84 since both 84 an 84/2 = 42 have spectral basis {21,28,36}, while index(84) = 1 and index(42) = 2.
%Y Cf. A330849, A335080, A335082, A335083, A335084, A335085.
%K nonn
%O 1,1
%A _Walter Kehowski_, May 24 2020