A335080 First elements of maximal isospectral chains of length 1, or, equivalently, numbers with spectral basis of index 1.

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

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: A168638 A024619 A361102 * A323304 A325411 A106543

Adjacent sequences: A335077 A335078 A335079 * A335081 A335082 A335083

Keyword

nonn

Author

Walter Kehowski, May 24 2020

Status

approved