login
First elements of maximal isospectral chains of length 4.
5

%I #4 Jun 19 2020 04:13:09

%S 488880,1525680,2870280,4930272,5890248,6374664,8862984,9658080,

%T 9739080,10338480,10544544,12719880,13985712,14777280,15543216,

%U 16109280,16293600,16682400,16747848,17722080,19376136,20822472,22178736,22842288,25517232,26056368,26927280

%N First elements of maximal isospectral chains of length 4.

%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) = 488880 since all four numbers, 488880/k, k=1..4, have spectral basis {91665, 108640, 97776, 69840, 120960}, while index(488880/k)=k, k=1..4.

%Y Cf. A330849, A335080, A335081, A335082, A335084, A335085.

%K nonn

%O 1,1

%A _Walter Kehowski_, May 24 2020