OFFSET
1,1
COMMENTS
Isospectral Chain Conjecture: There exist isospectral chains of any positive length.
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.
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.
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.
LINKS
Garret Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly, Vol. 108, No. 4 (April 2001), pp. 336-346.
Wikipedia, Idempotent (ring theory)
Wikipedia, Peirce decomposition
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn
AUTHOR
Walter Kehowski, May 24 2020
STATUS
approved