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) = 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.
CROSSREFS
KEYWORD
nonn
AUTHOR
Walter Kehowski, May 24 2020
STATUS
approved