%I #4 Jun 19 2020 04:13:03
%S 10980,35280,36180,43380,46980,47268,52164,59508,71604,73476,75780,
%T 87444,92880,94500,100980,101700,108180,122580,132480,139284,150948,
%U 151956,172980,176580,179172,198576,201168,202464,215424,235188,237384,237780,241380,245556
%N First elements of maximal isospectral chains of length 3.
%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) = 10980 since the three numbers 10980, 10980/2 = 5490, and 10980/3 = 3660 all have spectral basis {2745, 2440, 2196, 3600}, while index(10980) = 1, index(5490) = 2, and index(3660) = 3.
%Y Cf. A330849, A335080, A335081, A335083, A335084, A335085.
%K nonn
%O 1,1
%A _Walter Kehowski_, May 24 2020