A335085 First elements of maximal isospectral chains of length 6.

1400839158600, 2902429341000, 3949885485000, 9000942048000, 10563097053600, 13554828003600, 18867199233600, 26976351213000, 37127826792000, 42966550125000, 50742170640000, 54497942553600, 56675647917000, 191546420284800, 259917211125000, 294509464704000

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

Table of n, a(n) for n=1..16.

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

Cf. A330849, A335080, A335081, A335082, A335083, A335084.

Sequence in context: A172717 A258444 A288263 * A213601 A213646 A288276

Adjacent sequences: A335082 A335083 A335084 * A335086 A335087 A335088

Keyword

nonn

Author

Walter Kehowski, May 24 2020

Status

approved