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%I #4 Jun 19 2020 04:13:17
%S 5385063600,5978343600,6789558600,12965853600,31967238600,32035143600,
%T 37418554800,37884558600,44580472200,50221710000,69733758600,
%U 75900423600,77102532000,84093966000,85348494000,88147278000,89292423600,92472078600,98119278000,103449198600
%N First elements of maximal isospectral chains of length 5.
%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) = 5385063600 since all five numbers, 5385063600/k, k=1..5, have spectral basis {1009699425, 398893600, 861610176, 769294800, 850273200, 702399600, 792892800}, while index(5385063600/k)=k, k=1..5.
%Y Cf. A330849, A335080, A335081, A335082, A335083, A335085.
%K nonn
%O 1,1
%A _Walter Kehowski_, May 24 2020