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

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) = 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.
		

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, A335084.