A335084 First elements of maximal isospectral chains of length 5.

5385063600, 5978343600, 6789558600, 12965853600, 31967238600, 32035143600, 37418554800, 37884558600, 44580472200, 50221710000, 69733758600, 75900423600, 77102532000, 84093966000, 85348494000, 88147278000, 89292423600, 92472078600, 98119278000, 103449198600

Offset: 1

Author

Walter Kehowski, May 24 2020

Keywords

• nonn

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.

Examples

			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.
		

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

Crossrefs

Cf. A330849, A335080, A335081, A335082, A335083, A335085.