Unitary Perfect Number

[koh]

    Hi, Seqfans

    
    I considered about "Ultra" Unitary Perfect Number such that UnitarySigma^{k}(m)=2*m , 3<=k . 

    For k=1 , k=2 are Unitary Perfect Number and Super Unitary Perfect Number, and they exist already on OEIS.

    For k=3, I got these examples.

    %S A000001 3,27,221,297,2376,144976 

    For k=4, I got only one example up to 10^6.

    %S A000002 981376    

    I conjectured that for all k, an example exists.


    If the equation is involved with ordinary Sigma, then I suppose that no example exists.

    Sigma^{k}(m)=2*m , 3<=k . 

   
              
    I think the following conjecture is also possible.

         "For all k, the largest integer n exists which satisfies the equation : UnitarySigma^{k}(m)=n*m . " 

    k   largest n            largest example m           up to  
    1           2     146361946186458562560000         2*10^12
    2           4                           18          400000
    3           6                          780          400000
    4          12                            6          400000
    5          18                           10          400000

         Where the numbers of right side mean limits of exhaustive search.

    It seems that the largest n appears at rather small m, and for each k if m -> large then n -> small.

    %S A000003 2,4,6,12,18

    %S A000004 146361946186458562560000 ,18,780,6,10

    I think that the two sequences are correct but I have no proof.

    Yasutoshi