OFFSET
1,1
COMMENTS
Isospectral Chain Conjecture: There exist isospectral chains of any positive length.
A number N is the first element of the first isospectral chain of length n if there is no integer M < N such that M is also the first element of an isospectral chain of length n. Then a(n)=N, where N is the first element of the first isospectral chain of length 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
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} of index 1.
a(2) = 84 since 84 = 2*42 and both 84 and 42 have the spectral basis {21, 28, 36}, and 42 has index 2. Also, 84 is maximal since 84/3 = 28 has spectral basis {21, 8}.
a(3) = 10980 since 10980 = 2*5490 = 3*3660 and all three integers 10980, 5490, 3660 have spectral basis {2745, 2440, 2196, 3600}. Also, 10980 is maximal since 10980/4 = 2475 has spectral basis {2440, 2196, 855}.
a(4) = 488880 since 488880 = 2*244440 = 3*162960 = 4*122220 and all four integers 488880, 244440, 162960, 122220 have spectral basis {91665, 108640, 97776, 69840, 120960}. Also, 488880 is maximal since 488880/5 = 97776 has spectral basis {91665, 10864, 69840, 23184}.
a(5) = 5385063600 since 5385063600 = 2*2692531800 = 3*1795021200 = 4*1346265900 = 5*1077012720, and all five integers 5385063600, 2692531800, 1795021200, 1346265900, 1077012720 have spectral basis {1009699425, 398893600, 861610176, 769294800, 850273200, 702399600, 792892800}. Also, 5385063600 is maximal since 5385063600/6 = 897510600 has spectral basis {112188825, 398893600, 861610176, 769294800, 850273200, 702399600, 792892800}.
a(6) = 348751729800 since 348751729800 = 2*174375864900 = 3*116250576600 = 4*87187932450 = 5*69750345960 = 6*58125288300, and all six integers 348751729800, 174375864900, 116250576600, 87187932450, 69750345960, 58125288300 have spectral basis {43593966225, 38750192200, 41850207576, 55066062600, 56250279000, 56196961200, 57044061000}. Also, 348751729800 is maximal since 348751729800 is not divisible by 7.
a(7) = 1524738985849800 since 1524738985849800 = 2*762369492924900 = 3*508246328616600 = 4*381184746462450 = 5*304947797169960 = 6*254123164308300 = 7*217819855121400, and all seven integers 1524738985849800, 762369492924900, 508246328616600, 381184746462450, 304947797169960, 254123164308300, 217819855121400 have spectral basis {190592373231225, 169415442872200, 182968678301976, 186702732961200, 89690528579400, 196740514303200, 193276772854200, 208868354226000, 106483588520400}. Also, 1524738985849800 is maximal since 1524738985849800/8 = 190592373231225 has spectral basis {169415442872200, 182968678301976, 186702732961200, 89690528579400, 6148141071975, 2684399622975, 18275980994775, 106483588520400}.
CROSSREFS
KEYWORD
nonn
AUTHOR
Walter Kehowski, Feb 08 2020
STATUS
approved