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A335080 First elements of maximal isospectral chains of length 1, or, equivalently, numbers with spectral basis of index 1. 5
6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 72, 74, 75, 76, 77, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106 (list; graph; refs; listen; history; text; internal format)
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..64.

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) = 6 since 6 has spectral basis {3,4} and, since 3+4=1*6+1, index(6) = 1.

CROSSREFS

Cf. A330849, A335081, A335082, A335083, A335084, A335085.

Sequence in context: A064040 A168638 A024619 * A323304 A325411 A106543

Adjacent sequences:  A335077 A335078 A335079 * A335081 A335082 A335083

KEYWORD

nonn

AUTHOR

Walter Kehowski, May 24 2020

STATUS

approved

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Last modified April 17 03:44 EDT 2021. Contains 343059 sequences. (Running on oeis4.)