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%I #4 Jun 19 2020 04:13:23
%S 1400839158600,2902429341000,3949885485000,9000942048000,
%T 10563097053600,13554828003600,18867199233600,26976351213000,
%U 37127826792000,42966550125000,50742170640000,54497942553600,56675647917000,191546420284800,259917211125000,294509464704000
%N First elements of maximal isospectral chains of length 6.
%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) = 1400839158600 since all six numbers, 1400839158600/k, k=1..6, have spectral basis {175104894825, 184472646400, 224134265376, 200119879800, 227163106800, 179924295600, 209920069800}, while index(1400839158600/k)=k, k=1..6.
%Y Cf. A330849, A335080, A335081, A335082, A335083, A335084.
%K nonn
%O 1,1
%A _Walter Kehowski_, May 24 2020
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