[seqfan] m/n-Unitary SN

[zbi74583.boat at orange.zero.jp]

    Hi,Seqfans
    [1/m-Aliquot Sequence]

    Definiition
    a(n)=1/m*Sigma(a(n-1))
    Where  m is integer

    Three cases exist.
    1. Periodic
    2. It becomes non integer
    3. Divergent

    The first one is 1/m-Sociable Number of order k.
    a(n+k)=a(n), k is number of the members of a cycle.
    m-Multipry Perfect Number is the fixed point of mapping E0.

    A113546, A113791, A113285

    I found one more  example of 1/4-SN

    6943520030720, 5386209454080, 6389772480000, 7665533854902, 575659229184,
5657834203560, 6382469882880, 7938550287360, 9055131471360, 9256155068160,
11798498589696, 12316979136000, 13937317954560, 18314063179776,
18071888486400, 20136798781440, 18850467643392, 16492422758400,
15699547839573, 6635615891520, 7400619883008, 6494644316160, 8009835724800,
9317732578770, 7847237806128

    Could anyone confirm it?

    Recently I computed by hand the case of Unitary Sigma.

    [m/n-Unitary Aliquot Sequence]
    a(k)=m/n*UnitarySigma(a(k-1)) .... E1
    Where m,n are integer, m<n

    If it becomes periodic then it is m/n-Unitary Sociable Number.
    Note that m/n-Unitary RMPN is the fixed point of mapping E1.

    [Results]
    2/3-Unitary SN

    2^3, 2*3
    2^2*3*5, 2^4*5, 2^2*17

    3/4-Unitary SN

    3^3, 3*7, 2^3*3
    3^4*5, 3^2*41, 3^2*5*7, 2^3*3^2*5
    2^3*3^4*5, 3^4*41, 3^2*7*41, 2^3*3^2*5*7
    2^3*3^4*5*7, 2^3*3^4*41, 3^4*7*41, 2^3*3^2*7*41



    Yasutoshi